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UNIVERSITY   OF   CALIFORNIA 


DE 


No.  1 


)N 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/arithmeticforuppOOwalsrich 


AN   ARITHMETie-^ 


FOR  UPPER   GRADES 


BY 


JOHN   H.    WALSH 

ASSOCIATE    CITY    SUPKRINTENDEXT    OP    SCHOOLS 
THE    CITY    OF    NEW    YORK 


-oo>a<o<>- 


BOSTOX,   U.S.A. 

D.    C.    HEATH   &   CO.,   PUBLISHERS 

1908 


Copyright,  1908, 
Bt  JOHN  H.   WALSH. 

EDUCATION  O^rt. 


PREFACE. 

An  Arithmetic  for  Upper  Grades,  while  intended  chiefly 
for  pupils  of  the  last  two  years  of  the  elementary  school, 
has  been  arranged  to  include  the  work  of  the  sixth  grade. 

The  seventh  year  portion  comprises  a  simple,  but  S3^stem- 
atic,  treatment  of  commercial  arithmetic,  including  per- 
centage with  its  several  applications,  and"  elementary 
exercises  involving  the  employment  of  common  business 
forms.  This  is  preceded  by  reviews  of  fractions  and  deci- 
mals, and  is  followed  by  miscellaneous  problems,  oral  and 
written. 

The  eighth  year  section  contains  the  remaining  topics 
of  the  ordinary  course,  prefaced  by  a  review  of  compound 
denominate  numbers  and  simple  measurements.  The  mis- 
cellaneous problems  that  immediately  follow  are  limited 
to  the  subjects  treated  in  this  portion,  so  as  to  be  available 
for  use  in  such  schools  as  teach  the  seventh  and  eighth 
grades  in  combined  classes  alternating  the  work  of  each 
year. 

A  scientific  treatment  of  numbers  and  processes  is  next 
presented,  which  may  be  taken  up  at  any  stage.  The  mis- 
cellaneous reviews  that  follow  cover  all  the  ground  previ- 
ously studied. 

In  many  schools  whose  courses  of  study  require  an  ad- 
vanced text-book  in  the  sixth  year,  it  is  customary  to  begin 
the  arithmetic  work  of  this  grade  with  the  development  of 
formal  definitions,  principles,  rules,  etc.  In  this  case,  the 
section  on  numbers  and  processes  should  first  be  taken  up, 
then  the  fraction  and  decimal  reviewsj  followed  b}^  the 
reviews  of  compound  numbers  and  measurements. 


CONTENTS. 

CHAPTER   L 

PAGES 

Review  of  Fractions  ........  1-24 

Review  of  Fractions  ;  Type  Problems ;  Review  of  Deci- 
mals. 

CHAPTER    II. 
Percentage  .........  25-75 

Finding  the  Percentage,  the  Base,  and  the  Rate  ;  Amount 
and  Difference  ;  Profit  and  Loss  ;  Commission  and  Broker- 
age ;  Insurance ;  Taxes  and  Duties  ;  Stocks  and  Bonds ; 
Commercial  Discount. 

CHAPTER   III. 
Applications   of    Percentage    involving    the    Element   of 

Time 70-11.3 

Interest ;  Partial  Payments  ;  Compound  Interest ;  Annual 
Interest ;  Present  Worth  ;  To  find  the  Bank  Discount ;  Prob- 
lems in  Interest  and  Bank  Discount. 

CHAPTER   IV. 

Business  Forms  and  Usages;   Review      ....       114-140 
Transmission    of    Money ;     Bank    Drafts ;    Foreign    Ex- 
change ;  Bills  and  Accounts  ;  Miscellaneous  Drills. 

CHAPTER  V. 
Denominate  Numbers  ;  Measurements  ....  141-178 
Review  of  Denominate  Numbers  ;  Standard  Time  ;  Longi- 
tude and  Time  ;  Area  of  Rectangles  ;  Volume  of  Rectangular 
Solids ;  Board  Measure  ;  Masonry  and  Brickwork  ;  Painting 
and  Plastering ;  Roofing  and  Flooring ;  Carpeting  and 
Papering ;   The  Metric  System. 

V 


vi  Contents 

CHAPTER   VI. 

PAGES 

Ratio  and  Proportion  ;   Poweus  and  Roots    .         .         .      179-206 
Arithmetical  Analysis  ;  Analysis  by  Aliquot  Parts  ;  Ratio  ; 
Inverse  Ratio  ;  Proportion  ;  Partitive  Proportion  ;  Partner- 
ship ;  Powers  ;  Roots  ;  Applications  of  Powers  and  Roots. 

CHAPTER   VII. 
Mensuration  ;    Miscellaneous  Problems  .         .         .      207-245 

Areas  of  Plane  Surfaces ;  Area  of  a  Polygon  ;  Area  of  a 
Regular  Hexagon ;  United  States  Public  Lands ;  Circum- 
ference and  Area  of  a  Circle  ;  Prisms  ;  Cylinders  ;  Pyramids 
and  Cones ;  The  Sphere  ;  Miscellaneous  Problems. 

CHAPTER   VIII. 

General  Review         .         .         .         .         .         .         .         .      246-298 

Numbers  ;  Notation  and  Numeration  ;  Reductions ;  Fac- 
tors and  Multiples ;  The  Fundamental  Processes  :  Addition, 
Subtraction,  Multiplication,  Division  ;  Miscellaneous  Prob- 
lems. 


AN    ARITHMETIC    FOR   UPPER 
GRADES. 

CHAPTER   I. 
REVIEW   OF   FRACTIONS. 

1.  As  a  preliminary  to  the  regular  percentage  work  of 
the  seventh  year,  it  will  often  be  found  profitable  to  give  a 
short  time  to  the  review  of  fractions,  common  and  decimal. 
At  intervals  throughout  the  year,  a  few  minutes  of  an  arith- 
metic period  should  be  spent  in  rapid  oral  reviews,  employ- 
ing the  drills  and  the  sight  exercises  of  this  chapter. 

Drill  Exercises. 

Note.  The  fractions  in  the  answers  should  be  given  in  their  low- 
est terms. 

2.  Add: 


13. 


19. 


1 

3" 

2. 

2 
5 

3. 

4 
9 

4. 

2. 
5 

5. 

2 
9 

6. 

4 
9 

i 

1 

1 
"9 

1 

2 

4 
¥ 

2 
3 

8. 

3 

5 

9. 

5 
9 

10. 

2 
3 

11. 

4 

■5 

12. 

8 
9 

1 
3 

2 
5 

4 

"9 

2 
t 

3. 
5 

5 
9 

3i 

14. 

H 

15. 

6f 

16. 

02 

17. 

Pf 

18. 

8* 

1 
•   "3 

1 
"5 

1 

¥ 

* 

2 
¥ 

i 

5| 

20. 

6f 

21. 

^9 

22. 

6^ 

23. 

6i 

24. 

13| 

i 

2 
"5" 

4 

9 

2 
t 

3 
5 

5 

9 

-l.'  Arithmetic 

25.        4i  26.     of  27.      9|      28.     5?-  29.      8f  30.     6| 

4  6i  2i               6|  7|  9| 

31.         4|  32.     6f  33.      8f      34.      9J  35.      7f  36.      2| 

31  3|  4|              8|  6J  31 


37.  \      38.      4J      39.        f      40.    5yV     41.        f      42.     ^ 

1  1  .SI  Q-l-  3.  2. 

4  6"  ^8  ^12  8  9 


43. 


1. 


7.     8 


13. 


19. 


25. 


— 

9fV 
3A 

45. 

7 
8 
5 
8- 

46. 

^ 

7 
9" 

47. 

7 
1  0 

•^10 

48. 

7+i 

Subtract: 

1   2. 

1 

5" 

7 
9 
5 
9 

3. 

6| 

2 
5 

4. 

8^ 

^9 
4 
9" 

5. 

74 

6. 

6| 

^9 

8    8. 

6 

9. 

7 

10. 

4 

11. 

6 

12. 

8 

1 

2 
"5 

5 
9 

^^3 

H 

75 
'9 

8   14. 

6 

15. 

16. 

9 

17. 

■  6 

18. 

8 

^ 

If 

If 

3| 

4f 

or, 
^9 

81   20. 
1« 

6^ 
1* 

21. 

22. 

9i 
3| 

23. 

6| 
4f 

24. 

84 
^9 

^9 

f   26. 

1 

6 

7 
8 
5 
8 

27. 

7 
9 
4 
"9 

28. 

.5 
12 

29. 

Q5 

^8 
1 

30. 

88 

2 

"9 

4.  Multiply: 

1.  1X2            2.  3x1  3.  lof    4  4.  5x1 

5.  1  X  9           6.  17  X  1  7.  1  of  23  8.  31  X  yV 
9.  I  X  2          10.  3  X  f  11.  f  of    5  12.  .  7  X  I 


Review  of  Fractions 
Multiply : 


13. 

3i 

2 

14.    51 
3 

15. 

4 

16. 

5 

17.    81 
6 

18. 

2| 
2 

19.    5} 
3 

20. 

5 

21. 

Q2 

8 

22.    91 

7 

23. 

iof 

2 

24. 

3 

x* 

25. 

1 

8 

X    4 

26. 

6xi 

27. 

iof 

4 

28. 

6 

xi 

29. 

1 

8 

xl6 

30. 

27x1 

31. 

lof 

6 

32. 

9 

xi 

33. 

1 
8 

x20 

34. 

33x1 

35. 

fof 

2 

36. 

3 

x| 

37. 

■5 
6 

X    3 

38. 

7x^ 

39. 

foflo 

40. 

24 

xf 

41. 

.5 

8 

xl2 

42. 

3xj 

43. 

4i 
2 

44.    51 
4 

45. 

2 

46.    7| 
3 

47.    83I, 
6 

48. 

6i 
6 

49.    81 
9 

50. 

Q2 

3 

51.    4| 
4 

52.    6| 
5 

5. 

Divide : 

1. 

2)4  fifths 

2. 

3)6 

sevenths 

3 

.    4)8  ninths 

Note.  In  dividing  4  by  2,  the  pupil  may  think  2  into  4  fifths,  or 
i  of  4  fifths,  or  4  fifths  divided  by  2.  These  and  the  following  ex- 
amples are  placed  in  the  short-division  form  to  lead  pupils  to  refrain, 
at  times,  in  written  work  from  changing  the  mixed  number  in  the 
dividend  to  an  improper  fraction  when  the  divisor  is  a  whole  number. 

4-  7)14  5.  8)ii  6.  5)11  7.  6)i| 

8.  2}4|-  9.  3)6|-  10.  4}8|  11.  0)0^ 

12.  2)i  13.  3)1.  14.  4}i-  15.  5)i 

16.  2)121  17.  3)91  18.  4)16^  19.  5)151- 


Arith] 

raetic 

Divide : 

20.    6)18i 

21. 

7)211 

22. 

8)401 

23. 

9)63^ 

24.    5)1 

25. 

5)21 

26. 

m 

27. 

2)H 

28.   4)2| 

29. 

o)3f 

30. 

6)3| 

31. 

7)4| 

32.    2)191 

33. 

4)26i 

34. 

5)28| 

35. 

6)33f 

36.    2)i 

37. 

2)H 

38. 

311 

39. 

3)U 

40.    2)171 

41. 

3)161 

42. 

4)191 

43. 

5)421 

44.    6)431 

45. 

7)401 

46. 

8)411 

47. 

9)461 

6.   Preliminary  Exercises. 

1.  How  many  baseballs  at  $i  each  can  be  purchased 

for  |i ?     For  f  1  ?     For  | li  ? 

2.  1)1         3.    i)l  4.    1)11  5.    1)2 


7.    1)8 


8.    1)20         9.    1)6  10.    1)12 


How  is  the  quotient  obtained  in  each  case  ? 


12.    1)1         13.    Hlli        14.    1)1 


l^9i 

2. 


11.    1)12 


15.    11)3         16.    1^)6 


Multiply  the  divisor  and  the  dividend  in  each  of  the  five  preceding 
examples  by  2  : 


17.   3)3         18.    3)3^         19.    3)^         20.    3)^ 

How  do  the  quotients  compare  in  each  case  ? 


21.    3)12 


7.    1. 


Proof. 


Divide  12  by  If 

2.    Divide  21  by  |. 

U)12 

1)21 

x2    x2 

x4   x4 

3  )24 

3}84 

8  Ans. 

28  Ans. 

11  X  8=  12. 

Proof.     28  x  |  =  21. 

Review  of  Fractions  5 

Note.  In  mental  work  it  is  often  convenient  to  change  a  fractional 
divisor  to  a  whole  number  by  multiplying  the  divisor  by  the  denomi- 
nator of  the  fraction,  the  dividend  being  multiplied  by  the  same  num- 
ber.    Divide  the  new  dividend  by  the  new  divisor. 


8. 

Drill  Exercises, 
/^ide : 

1. 

li)9 

6.   21)15 

11. 

1)12 

16. 

nn 

21. 

f)2i 

2. 

H)15 

7.   31)21 

12. 

1)12 

17. 

^)H 

22. 

I)2| 

3. 

11)18 

8.    2^)27 

13. 

*)12 

18. 

H)H 

23. 

im 

4. 

H)15 

9.   3i)26 

14. 

f)io 

19. 

^)^ 

24. 

iM 

5. 

4)20 

10.   1|)15 

15. 

*)14 

20. 

l|)16i 

25. 

f)6i 

9.  Oral  Problems. 

1.  A  farmer  sold  15^  cords  of  wood  in  January  and  10|^ 
cords  in  February.     How  many  cords  did  he  sell  in  all  ? 

2.  From  a  piece  of  cloth  containing  30  yards,  121  yards 
are  sold.     How  many  yards  remain  ? 

3.  A  rectangular  field  is  12^  rods  long  and  7^  rods  wide. 
How  many  rods  of  fence  will  be  needed  to  inclose  it  ? 

4.  How   many  i-pound   packages  will   24 J   pounds   of 
candy  make  ? 

5.  A  traveler  walked  60J  miles  in  3  days.     How  many 
miles  a  day  did  he  average  ? 

6.  How  many  square  rods  are  there  in  a  field  20|^  rods 
long  and  10  rods  wide  ? 

7.  Mr.  Yates  pays  $17}  for  carpet  and  $20}  for  furni- 
ture.    What  is  the  amount  of  his  bill  ? 

8.  How  many  minutes  are  there  in  i  of  a  day  ? 

9.  At  60  pounds  per  bushel,  what  will  J  bushel  weigh  ? 

10.  How  many  yards  of  cloth  at  $  li  per  yard  can  be 
bought  for  $12? 


6  Arithmetic 

10.   "Written  Problems. 

1.  A  boy  sold  16|  dozen  eggs  at  one  time  and  20 1  dozen 
at  another  time.     How  many  eggs  did  he  sell  ? 

2.  Find  the  sum  of  four  numbers,  two  of  which  are  15j\ 
and  19j^5,  respectively,  the  third  being  equal  to  the  sum  of 
these  two,  and  the  fourth  being  equal  to  their  difference. 

3.  Two  trains  start  from  the  same  point  and  move  in 
opposite  directions,  each  at  the  rate  of  32i  miles  per  hour. 
How  far  apart  are  they  in  4  hours  ?     . 

4.  What  is  the  total  weight  of  16  barrels  of  sugar, 
averaging  310|^  pounds  each  ? 

5.  A  crop  of  wheat  averaged  12i  bushels  per  acre. 
How  many  acres  were  required  to  produce  500  bushels  ? 

6.  How  many  square  rods  are  there  in  a  rectangular  field 
160^  rods  by  84  rods  ? 

7.  A  train  starting  at  10.45  a.m.  reaches  a  town  140 
miles  distant  at  2.15  p.m.  How  many  miles  per  hour  does 
it  average  ? 

8.  If  3  eighths  of  a  number  is  147,  what  is  1  eighth  of 
the  number  ?     What  is  the  number  ? 

9.  A  rectangular  lot  is  120  feet  long.  Its  width  is  y^^ 
of  its  length.  How  many  running  feet  of  fence  will  be 
required  to  inclose  it  ?     (Make  a  diagram.) 

10.  How  many  gallons  are  there  in  IJ  barrels  of  31  ^  gal- 
lons each  ? 

11.  Sight  Exercises.  » 

Note.  To  accustom  the  pupils  to  avoid  unnecessary  figures,  fre- 
quent drills  in  sight  and  blackboard  exercises  are  important.  Pupils 
should  give  orally  the  answers  to  the  following  examples,  or  should 
promptly  write  the  answer  to  each  at  a  signal,  the  pupil  being  ex- 
pected to  know  the  answer  before  beginning  to  write. 


Review  of  Fractions  7 


Add: 


1.   241         4.   17|  7.   48f 

3f  51  31 


42|         5.    81  8.   84i| 

81  36|  91 


3.       931  6.  3f  9.       46| 

_1j  ^  3 

12.    Blackboard  Exercises. 

Note.  Pupils  are  expected  to  write  only  the  answers  to  the  follow- 
ing examples,  but  time  should  be  allowed  them  to  write  the  total  of 
each  column  as  they  obtain  it.  These  exercises  are  designed  to  show 
pupils  that  it  is  not  always  necessary  to  rewrite  the  fractions  with  a 
common  denominator. 


Add : 


241  4.  40^  7.  471 

6^  281  73. 

8.  461 


2.    47| 

5.   48J 

181 

4 

32i 
7f 

3.    841 

loi 

6.   23^ 
45| 

3tV 

6i 

13.   Sight  Exercises, 

Subtract : 

1.    18f 

3.    721 

64 

2i 

30| 
411 

5tV 


5.  40^ 


41 


2.  541  4.  271  6.  801 

31  53  71 

"^3  ^¥  '3^ 


Arithmetic 


Subtract : 


7. 


361 


8.     62,1, 
81 


14.   Blackboard  Exercises. 

Subtract : 

1.      841 
291 

4. 

401 

16| 

2.      90^^ 
261 

5. 

60| 
23| 

3.      78/^ 
39i 

.6. 

181 

9. 


9. 


931 
101 


631 

52tV 

347 


93| 
471 


15.    Sight  Exercises. 
Multiply : 


2. 


8 


20| 


21J 


4.  16 

41 

5.  24 

_2i 

6.  48 

14- 


8. 


101 
12 

12f 


40f 


16.   Blackboard  Exercises. 

Multiply : 

1.  1241  4. 

7 


2.  320f 


3.  621f 


6. 


304f 
12 


5.  4234 
2 


516f 


10 


8. 

22233, 
13 

9. 

2011 
14 

Review   of  Fractions  9 

17.  Sight  Exercises. 

Divide : 

1.  2)_26i  4.    7)781  7.   6)671 

2.  3)39|  5.   8)17f  8.   5)51| 

3.  4)364.  6.    9)36^  -9.   4)27f 

18.  Blackboard  Exercises. 
Divide : 

1.  2)2461  4.    5)8491  7.    8)649^ 

2.  3)4591        5.  6)2731         8.  9)833| 

3.  4)7231        6.  7)723|         9.  10)537| 

19.  Written  Exercises. 

Note.     Determine  the  common  denominator  by  inspection. 
Find  results : 

1.  81  + 7L 4-131  +  421    3.  28f +  45i  +  83|  +  96i 

2.  26J  +  30f  + 471 +  56|   4.  35f  +  563-V  +  971  +  481  f 
5.  19if  +  12|  +  24||  +  87| 

9.    862|-258| 

10.  683y^-423if 

11.  7091-357^% 

To  multij:)!!/  two  mixed  numbers,  reduce  them  to  improj^er 
fractions,  multiply  the  numerators  together  and  the  denomina- 
tors together,  and  reduce  the  resulting  fraction,  if  possible. 

12.  4fx4f  15.    15fxl0| 

13.  2^x6i  '        16.   171x5^ 

14.  3ix5|  17.    12fx7i 


6. 

910|- 

.316j\ 

7. 

862 1 - 

-2o8f 

8. 

200tV 

-103^V 

18. 

4f^4| 

19. 

71^3,^ 

20. 

5f-3t 

21. 

16i-33i 

22. 

20|^41f 

lO  Arithmetic 

To  divide  by  a  /inaction,  multiply  by  the  divisor  inverted. 

23.    (3f  +  5t)x4| 

24.  il^-^)-^^ 

25.  6|x(17f  +  9f) 

26.  (66J -36f)-f-2f 

27.  (3fx5|)-(7f^3/,) 

TYPE   PROBLEMS. 
MULTIPLICATION  OR  DIVISION:    ONE  OPERATION. 

20.  Preliminary  Exercises. 

1.  At  12  cents  per  yard,  find  the  cost  of  2  yards  of  dress 
goods.     Of  2|  yards.     Of  \  yard.     Of  f  yard. 

To  indicate  the  operation  required  in  each  case,  the  sign  of  multipli- 
cation is  employed:  12;z^  x  2,  12^  X  2i,  12;^^  x  |,  12;*  x  |. 

2.  Find  the  price  per  yard  when  2  yards  cost  24  cents. 
When  2\  yards  cost  30  cents.  When  ^  yard  costs  6  cents. 
When  I  yard  costs  9  cents. 

In  each  of  these  examples  the  price  per  yard  is  obtained  by  dividing 
the  total  cost  by  the  number  of  yards:  24jz^ -=- 2,  30^-4-2^,  6^^^, 
9^-1. 

21.  Oral  Problems. 

Note.  In  solving  each  of  the  following  problems,  the  pupils  should 
first  state  whether  it  is  an  example  in  multiplication  or  in  division. 
They  may  easily  determine  this  by  mentally  substituting  a  whole 
number  for  the  traction. 

1.  A  24-acre  field  is  divided  into  plots  of  |  acre  each. 
How  many  plots  are  there  ? 

2.  At  $J  per  bushel,  find  the  cost  of  56  bushels  of 
wheat. 

3.  How  many  cords  of  wood  in  32  piles  containing  | 
cord  each? 


Review  —  Type   Problems  1 1 

4.  If  a  train  goes  f  mile  in  a  minute,  how  many  minutes 
will  it  take  to  go  60  miles? 

5.  A  dealer's  profit  is  J  of  the  cost.  What  is  the  cost, 
if  his  profit  is  $24? 

6.  How  many  |-pound  packages  can  be  filled  from  a 
36-pound  box  of  tea  ? 

7.  A  drover  sells  |  of  his  herd  of  120  cattle.  How 
many  does  he  sell  ? 

8.  Nine  tenths  of  the  pupils  of  a  certain  class  are  pres- 
ent. There  are  27  present.  How  many  pupils  belong  to 
the  class  ? 

9.  If  a  man  can  do  two  fifths  of  a  piece  of  work  in  a 
day,  how  long  will  it  take  him  to  do  the  whole  work  ? 

Number  of  days  =  1  work  h-  |  work  =  f  work  -^  |  work  =  5-7-2. 

10.    How  long  will  it  take  a  pipe  discharging  |  gallon  per 
second  to  empty  a  tank  containing  60  gallons  ? 

22.   Written  Problems. 

Note.     Before  solving  the  following  problems,  the  required  opera- 
tion should  be  indicated  in  each  case  by  the  use  of  the  proper  sign. 

1.  Into  how  many  building  sites  of  f  acre  each  can  a 
farm  of  192  acres  be  divided  ? 

Number  of  sites  =  192  A.  -4-  |  A. 

2.  Find  the  cost  of  784  bushels  of  wheat  at  $  jf  per 
bushel. 

Cost  =  $  i|  X  784. 

3.  How  many  loads,  each  containing  ^  cord,  are  there  in 
336  cords  of  wood  ? 

4.  What  time  will  it  take  a  train  to  go  195  miles  at  the 
rate  of  |  mile  a  minute  ? 

5.  At  95^  per  bushel,  how  many  bushels  of  wheat  can 
be  bought  for  $  142.50  ? 


12  Arithmetic 


6.  How  many  bushels  of  wheat  at  $  i|  per  bushel  can 
be  bought  for  $  142i  ? 

7.  If  it  takes  J  yard  of  material  to  make  an  apron,  how 
many  yards  will  be  required  to  make  144  aprons  ? 

8.  How  many  vests  can  be  made  from  144  yards  of 
cloth,  if  f  yard  is  needed  for  each  ? 

9.  If  three  men  working  together  can  do  2V  +  2^5  +  3V 
of  a  piece  of  work  in  a  day,  how  long  will  they  require  to 
do  the  whole  work  ? 

10.  Find  the  cost  of  if  acre  of  land  at  $  256  per  acre. 

11.  H  a  horse  eats  |  bale  of  hay  in  a  week,  how  long 
will  a  bale  last  ?    32  bales  ? 

12.  A  farmer  sold  his  farm  for  |  of  its  cost,  which  was 
14800.     What  did  he  receive  for  it  ? 

13.  A  can  do  J  as  much  work  in  a  day  as  B.  How  many 
days  would  he  require  to  do  a  piece  of  work  that  B  could 
finish  in  105  days  ? 

14.  A  and  B  together  can  do  y-  as  much  work  as  B  alone. 
How  many  days  would  both  working  together  require  to  do 
a  piece  of  work  which  B  can  do  in  105  days  ? 

15.  A  dealer's  profits  average  jVo  of  the  cost  of  the  goods 
sold.     How  much  does  he  gain  on  goods  costing  $  275  ? 

16.  If  the  weight  of  roasted  coffee  is  |^  of  the  weight  of 
unroasted  coffee,  how  many  pounds  of  the  latter  will  be  re- 
quired to  make  221  pounds  of  roasted  coffee  ? 

Suggestion.  In  this  problem  and  in  the  remaining  four,  the  pupil 
may  use  x  as  follows  : 

Let  X  represent  the  number  of  pounds  of  unroasted  coffee. 
Then,  ^ia;  =  221 

a:  =  221  --  H,  etc. 

17.  A  certain  number  multiplied  by  J  gives  115^^  as  the 
result.     What  is  the  number  ? 


Review  —  Multiplication  and   Division        13 

18.  What  must  be  the  capacity  of  a  bin  in  cubic  feet  to 
hold  385  bushels  of  grain,  assuming  4  bushel  to  a  cubic 
foot? 

19.  A  man  sold  articles  for  ij  of  the  cost,  receiving  for 
them  $  255.     What  was  the  cost  ? 

20.  Five  ninths  is  |  of  what  fraction  ? 

MULTIPLICATION  AND   DIVISION. 

23.    Oral  Problems. 
Unitary  Analysis. 

1.  If  2  yards  of  calico  cost  16  cents,  what  will  3  yards 
cost  ? 

First  find  the  cost  of  1  yard. 

2.  Find  the  cost  of  2i  yards  of  dress  goods  at  the  rate 
of  40  cents  for  4  yards. 

3.  If  5  men  require  40  days  to  do  a  piece  of  work,  how 
long  would  it  take  8  men  to  do  it  ? 

4.  A  can  mow  J  of  a  field  in  9  days.  How  many  days 
would  he  require  to  mow  |  of  it  ? 

5.  If  it  requires  160  rods  of  wire  for  a  fence  4  strands 
high,  how  many  rods  would  be  needed  for  a  5-strand  fence 
of  the  same  length  ? 

6.  If  a  man  can  walk  20  miles  in  5  hours,  in  how  many 
hours  can  he  walk  12  miles  ? 

7.  To  paint  a  house  requires  4  men  12  days.  How  long 
would  it  take  6  men  ? 

8.  A  train  goes  16  miles  in  30  minutes.  How  many  miles 
will  it  go  in  li  hours  ? 

9.  To  build  a  bridge  required  the  labor  of  10  men  for 
24  days.     How  many  men  could  complete  it  in  16  days  ? 

10.    If  12  acres  produce  36  tons  of  hay,  how  many  tons 
will  32  acres  produce  at  the  same  rate  ? 


14  Arithmetic 

Ratio  Method. 

11.  At  the  rate  of  75  cents  per  dozen  bunches,  what  will 
be  the  cost  of  4  bunches  of  rhubarb  ? 

4  is  ^  of  a  dozen. 

12.  If  a  certain  amount  of  hay  will  last  14  horses  4^- 
months,  how  many  horses  will  eat  it  in  li  months  ? 

41  months  is  3  times  1|  months. 

13.  What  will  be  the  cost  of  13  pounds  of  coffee  at  the  rate 
of  $  27.90  for  a  bag  of  130  pounds  ? 

14.  If  a  60-foot  rail  weighs  560  pounds,  what  will  be  the 
weight  of  a  piece  15  feet  long  ? 

15.  If  31  acres  produce  400  bushels  of  wheat,  what  will 
be  the  yield  of  93  acres  at  the  same  rate? 

16.  Seven  cords  of  beech  produce  as  much  heat  as  9  cords 
of  pine.  How  many  cords  of  pine  will  produce  as  much 
heat  as  35  cords  of  beech  ? 

17.  If  24  reams  of  paper  are  used  in  printing  900  copies 
of  a  book,  how  many  reams  will  be  required  for  300 
copies  ? 

18.  A  farmer  finds  that  he  has  obtained  82  bushels  of 
corn  from  30  shocks.  What  will  be  the  yield  from  90 
shocks  at  the  same  rate? 

19.  If  9  cords  of  wood  are  required  to  make  8  tons  of 
paper,  how  many  cords  will  be  required  to  make  72  tons  ? 

20.  Mr.  Freeman  paid  $  10,000  for  a  farm  of  160  acres. 
He  agrees  to  sell  a  portion  of  it  at  the  rate  paid  for  the 
whole.     What  should  he  receive  for  32  acres  ? 

24.   "Written  Problems. 

1.    If  28f  acres  yield  a  profit  of  $230,  what  will  be  the 
profit  on  37 J  acres  at  the  same  rate? 


Review — Multiplication  and   Division        15 

28f  A.  yield         $230. 

1  A.  yields       ^^. 

28| 

37iA.yieldt23^^<^. 

^        "^  28| 

Reducing  the  mixed  numbers  to  improper  fractions,  we  have, 

M30jiiAl5.     Cancel. 
115  X  2 

2.    If  16  men  require  311  days  to  do  a  piece  of  work, 

how  long  will  it  take  28  men  to  do  it  ? 

16  men  require         31 1  da. 

1  man  requires  31^  da.  x  16. 

311  da.  X  16 


28  men  require 


28 


Number  of  days  = Cancel. 

2  X  28 

3.  If  4^  times  a  certain  number  is  221,  what  is  12f  times 
the  same  number  ? 

4.  Find  the  cost  of  9  pairs  of  stockings  at  the  rate  of 
$2.80  per  dozen  pairs. 

5.  How  many  bushels  of  oats  at  32  pounds  per  bushel 
will  be  equal  in  weight  to  2400  bushels  of  corn  weighing 
56  pounds  per  bushel  ? 

6.  What  is  the  value  of  579  (German)  marks  in  (French) 
francs,  the  latter  being  worth  19^^  cents  in  United  States 
money,  and  the  former  234  cents  ? 

7.  If  $  600  yield  $  30  interest  in  a  year,  how  much 
interest  should  $  720  yield  in  the  same  time  ? 

8.  What  will  be  the  cost  of  7000  pounds  of  coal  at 
f  4.48  per  long  ton  of  2240  pounds  ? 

9.  A  train  requires  18  hours  (running  time)  to  cover  a 
certain  distance  when  going  at  the  rate  of  24|-  miles  per 
hour.    How  long  will  it  take  if  it  travels  30 J  miles  per  hour? 

10.    If  J  of  an  acre  of  land  shows  a  profit  of  $  15.80,  what 
is  the  profit  on  2J  acres  at  the  same  rate  ? 


1 6  .  Arithmetic 

25.  Oral  Exercises. 

1.  What  is  1  of  12?     |  of  24?     fof48?     J  of  100? 

2.  Six  is  J  of  what  number?  (6  =  ^ x.)  8  is  J  of  what  ? 
12  is  i  of  what?     20  is  \  of  what? 

3.  Sixteen  is  ^  of  what  number?  16  is  |  of  what? 
36  is  i  of  what?  36  is  i  of  what  ?  36  is  |  of  what?  36  is 
I  of  what?  60  is  i  of  what?  60  is  |  of  what?  60  is  | 
of  what?     60isf  of  what? 

4.  Twelve  is  what  part  of  24?  8  is  what  part  of  24? 
16  is  what  part  of  24  ?  6  is  what  part  of  24  ?  18  is  what 
part  of  24  ?     4  is  what  part  of  24  ?     20  is  what  part  of  24  ? 

5.  One  fourth  is  what  part  of  i?  i  is  what  part  of  i? 
i  is  what  part  of  i?  i  is  what  part  of  i?  J^  is  what  part 
ofi? 

26.  Oral  Problems. 

1.  A  farmer  sold  f  of  a  flock  of  72  sheep.  How  many 
did  he  retain? 

2.  Two  thirds  of  A's  farm  is  under  cultivation.  The 
remainder  of  the  farm  contains  75  acres.  How  many  acres 
are  under  cultivation  ? 

3.  A  man  can  do  f  of  a  piece  of  work  in  a  day.  How 
many  days  will  it  take  him  to  do  a  piece  thrice  as  great? 

4.  I  insure  my  house  for  4  of  its  value,  or  $  1600.  What 
is  the  value  of  my  house? 

5.  One  ninth  of^  the  pupils  of  a  certain  class  were  absent 
on  a  stormy  day.  Twenty-four  were  present.  How  many 
pupils  belonged  to  the  class  ? 

6.  Mr.  Jonas  raised  600  bushels  of  wheat.  He  sold  450 
bushels.  What  part  of  his  crop  did  he  sell?  What  part 
did  he  retain  ? 


Review — Multiplication  and   Division        17 

7.  In  an  orchard,  three  quarters  of  the  trees  are  apple 
trees,  and  the  remaining  90  are  cherry  trees.  How  many- 
apple  trees  in  the  orchard? 

8.  After  traveling  l  of  his  journey,  a  passenger  has  still 
to  go  240  miles.     What  distance  has  he  already  traveled? 

9.  Seven  ninths  of  the  cargo  of  a  vessel  consists  of  630 
tons  of  wheat.     How  many  tons  of  cargo  in  the  vessel? 

10.  If  silk  sells  for  $  J  per  yard,  how  much  can  be  bought 
for  a  dollar? 

11.  What  fraction  of  an  hour  is  50  minutes? 

12.  Change  |-  day  to  hours. 

13.  A  sold  a  horse  at  a  profit  of  i  of  its  cost,  receiving  for 
it  $  180.     What  did  it  cost  ? 

14.  The  population  of  a  village  has  decreased  -^^  in  a 
year.  What  was  the  population  a  year  ago,  if  the  present 
population  is  450  ? 

15.  A  boy  of  10  is  \  as  old  as  his  father.  The  former's 
age  will  be  what  fraction  of  the  latter's  in  10  years  ? 

27.    Written  Problems.  ' 

1.  A  drover  sold  y^  of  a  drove  of  484  sheep.  How  many 
sheep  did  he  retain  ? 

2.  Four  sevenths  of  a  plantation  is  under  cultivation. 
The  remainder  of  the  plantation  contains  375  acres.  How 
many  acres  are  there  in  the  plantation  ? 

3.  Three  men  can  do  i,  ^,  and  |,  respectively,  of  a  certain 
piece  of  work  in  a  day.  What  part  of  the  work  can  the 
three  do  together  in  a  day  ?  How  long  will  it  take  the  three, 
working  together,  to  do  the  whole  work  ? 

4.  My  house  is  damaged  by  fire  to  the  extent  of  j\  of  its 
value.  The  damage  amounts  to  $760.  What  was  the 
original  value  of  the  house  ? 


1 8  Arithmetic 

*5.  Three  seventeenths  of  the  pupils  of  a  school  were 
absent.  The  number  present  was  168.  How  many  pupils 
belonged  to  the  school  ? 

6.  Of  a  crop  of  594  bushels  of  corn  451  bushels  were 
sold.     What  part  of  the  crop  was  sold? 

7.  One  third  of  the  trees  in  an  orchard  are  apple  trees, 
two  thirds  of  the  remainder  are  peach  trees,  and  the  rest 
are  cherry  trees.  There  are  76  cherry  trees.  How  many 
apple  trees  in  the  orchard? 

8.  A  man  has  gone  f  of  his  journey  by  boat,  and  225 
miles  by  rail.  He  has  ^\  of  the  distance  yet  to  travel. 
What  is  the  length  of  his  trip? 

9.  A  vessel's  cargo  consists  of  405  tons  of  wheat,  270 
tons  of  ore,  and  the  remainder  of  hay,  the  last  being 
one  sixth  of  the  cargo.     How  many  tons  of  hay  are  there  ? 

10.   If  velvet  is  worth  $3|  per  yard,  what  part  of  a  yard 
can  be  purchased  for  75  cents  ? 

REVIEW  OF  DECIMALS. 

Sight  Exercises. 

28.  Reduce  to  common  fractions : 

1.  .5  5.  .25  9.  .125 

2.  .6  6.  .08  10.  .16 

3.  .60  7.  .625  11.  .36 

4.  .84  8.  .002  12.  .875 

29.  Reduce  to  decimals: 
X.   4  o.    g  ».    2ir 

9       JL  ft      -2-  10      1 

3.  2^7  7.    f  11.    -i^ 

4.  f  8.    \\  12.    I 


4 


13. 

.375 

14. 

.155 

15. 

.024 

16. 

.075 

13. 

f 

14. 

A 

15. 

^ 

16. 

A 

Review  of  Decimals 


19 


30.    Multiply: 
1.   300  X. 05 


2.   484  X  .25 


Note.     When  one  of  the  factors  is  .25,  .126,  etc.,  the  equivalent 
common  fraction  should  generally  be  employed. 


3. 

200  X  .06 

9. 

.375  X  80 

15. 

444 

X.25 

4. 

250  X  .4 

10. 

.25  X  84 

16. 

848  X  .5 

5. 

300  X .022 

11. 

.125  X  16 

17. 

48  X  .625 

6. 

500  X  .12 

12. 

.75  X  36 

18. 

24  X  .875 

7. 

120  X .24 

13. 

1.5  X  18 

19. 

2.4  X  .875 

8. 

150  X  .24 

14. 

.15  X  18 

20. 

.48  X  .625 

31. 

Divide : 

1. 

12  -  .03 

2.    24 

-.6 

12  -  .03  =  1200  -f-  3. 

Why  ?               24  - 

-  .6  =  240  -  6. 

3. 

22-25 

4.   15 

-24 

5. 

22  *:  25  =  ^^  =    ^  . 
25      100 

1^  -  .25 

15- 

24      8      1000 

12  -  .25  =  12  -  1 

=  12 

x4. 

6. 

33  -f-  .375 

33  -  .375  =  33-1 

=  Si 

5  X  1  =  11  X  8. 

7. 

8^.02 

12. 

15 --50 

17. 

11- 

-.25 

8. 

18  -  .06 

13. 

24 --32 

18. 

22- 

-.25 

9. 

24 -.4 

14. 

16^64 

19. 

36- 

-.75 

10. 

88  -  .022 

15. 

18 --20 

20. 

21- 

-.125 

11. 

54  -  .12 

16. 

30  -- 100 

21. 

12- 

-.375 

Written  Exercises. 

32.   Add: 

Note.     Change  common  fractions  to  decimals. 

1.    164 -h  84.7 -F  96/3 -h75ff 

2-    -3  +  f  J  +  7yj3  ^-  35.309  +  .0483 


20  Arithmetic 

Add: 

3.  S^Vo  +  .0087  +  35.348  +  .0907  +  ^ 

4.  .945  +  34.8  +  9.48  +  tVV  4- 826 

33.  Subtract: 

1.  15.36 -.89  7.       6.51-3.429 

2.  18.45-9.7  *  8.       25.2-7.625 

3.  126.344-85.9  9.   55.007-4.26 

4.  .90072 -.086  10.       89.4-83.576 

5.  17.09-3.25  11.   103.05-8.9306 

6.  809.72-48.4  12.  .6  -  .4725 

34.  Multiply: 
1.   46.78  by  .93 

Note,  In  multiplication  of  decimals,  as  well  as  in  all  other  exam- 
ples, a  pupil  should  estimate  the  probable  answer  before  performing 
the  operation.  The  answer  to  the  foregoing  should  be  less  than  46, 
for  .93  is  less  than  1. 

The  number  of  decimal  places  in  the  product  is  equal  to  the 
niimher  in  the  m\dtip)licand  added  to  the  number  in  the  multi- 
plier. 


2. 

9.76  X  15.4 

9. 

7.37  X  .648 

3. 

37.68  X  2.234 

10. 

47  X  .8  X  .2 

4. 

.98  X  .98 

11. 

.126  X  48 

5. 

30.75  X  .46 

12. 

33.343  X  2.95 

6. 

7.08  X  .096 

13. 

8.053  X  1.47 

7. 

.09  X  4.56 

14. 

76.2  X  4.86 

8. 

.0364  X .82 

15. 

14.9  X  .83 

Review  of  Decimals  2i 

35.    1.    Divide  217.32  by  .6. 

The  divisor  .6  is  changed  to  6  by  multiplying  it  by  10.  To 
multiply  217.32  by  10,  the  decimal  point  is  moved  one  place  to  the 
right,  which  makes  it  2173.2. 

6)2173.2 

362.2     Ans. 

To  divide  by  a  decimal,  make  the  divisor  a  whole  number 
by  removing  the  decimal  jjoint,  and  make  a  coii'esjwnding 
change  in  the  dividend.  The  number  of  decimcd  places  in 
the  quotient  will  be  equal  to  the  number  in  the  dividend  as 
changed. 

2.    Divide  1301.3  by  .077. 

77  thousandths  is  changed  to  a  whole  number  by  multiplying  by 
1000,  which  moves  the  decimal  point  3  places  to  the  right.  To  make 
the  same  change  in  the  dividend,  two  ciphers  must  be  annexed,  mak- 
ing the  new  dividend  1301300. 

16000    Ans. 


77)1301300 

77 

531 

462 
693 
693 


3.    Divide  3.576  by  800. 

The  divisor  in  this  case  should  be  changed  to  8  by  dividing  it  by 
100.  A  corresponding  change  in  the  dividend  will  be  made  by  mov- 
ing the  decimal  point  two  places  to  the  left. 


Ans. 

5.  .21504-9.6 

6.  21.504-240 

7.  3.2 -.064 

8.  .432  H-  .012 


. 

8). 03576 

.00447 

36. 

Divide : 

1. 

384  -  3.2 

2. 

2304  -  .48 

3. 

53.95  -  83 

4. 

53.95  -  .65 

22 


I 

Arithmetic 

Divide : 

9.     2.436  --  5.8 

15. 

6.5772-8.4 

10.       68.6 -.049 

16. 

511.344  -  .67 

11.         2.4 -.096 

17. 

734.4  -  .6 

12.        180-1.5 

18. 

371.448  -  .77 

13.      142.4  -  .4 

19. 

18.612  -  .09 

14.   105.28-5.6 

20. 

38.76  -  3.4 

37.    Oral  Problems. 
1.    At  $  .125  each,  what  will  be  the  cost  of  88  books  ? 


iix  88 


2.  A  rectangular  field  is  8.8  rods  long  and  25  rods  wide. 
How  many  square  rods  does  it  contain  ? 

Number  of  square  rods  in  area  =  number  of  rods  in  length  x  num- 
ber of  rods  in  width. 

3.  What  decimal  part  of  32  is  8  ? 

4.  How  many  hours  are  there  in  .625  day? 

5.  How  many  nails  can  be  made  from  a  piece  of  wire 
125  inches  in  length,  if  each  nail  requires  2.5  inches  ? 

6.  At  2.15  pounds  per  quart,  what  is  the  weight  of  10 
gallons  of  milk  ? 

7.  The  quotient  is  2.31,  the  divisor  is  .3.    What  is  the 
dividend  ? 

8.  If  wool  costing  20/  per  pound  loses  .2  of  its  weight 
in  cleaning,  what  is  the  cost  per  pound  of  the  cleaned  wool? 

9.  A  has  .75  as  many  sheep  as  B.     The  former  has  24 
sheep.     How  many  has  B? 

10.  What  will  be  the  cost  of  40,000  bricks  at  $6.50  per 
thousand  ? 

11.  A  farmer  sold  .75  of  a  flock  of  36  sheep.     How  many 
sheep  did  he  retain? 


Review  of  Decimals  23 

12.  Six  sevenths  of  a  farm  is  under  cultivation.  There 
are  120  acres  not  under  cultivation.  How  many  acres  are 
there  in  the  farm? 

13.  If  a  man  can  do  .125  of  a  piece  of  work  in  a  day, 
how  many  days  will  it  take  him  to  do  the  whole  work  ? 

14.  My  house  is  insured  for  $  1600,  which  is  .8  of  its 
value.     What  is  the  value  of  the  house? 

15.  Of  the  pupils  of  a  certain  class  .125  are  absent. 
There  are  21  present.    How  many  pupils  belong  to  the  class  ? 

16.  Seventy-five  hundredths  of  the  trees  in  an  orchard  are 
apple  trees,  and  the  others  are  cherry  trees.  If  there  are 
90  of  the  latter,  how  many  apple  trees  are  there  in  the 
orchard  ? 

17.  After  selling  .16  of  his  Oats,  Mr.  Davis  had  168 
bushels  remaining.     How  many  bushels  had  he  at  first  ? 

18.  What  decimal  part  of  a  day  is  15  hours? 

19.  Change  .4  ton  to  pounds. 

38.    "Written  Problems. 

1.  If  goods  costing  $268  are  sold  at  a  gain  of  .25  of 
the  cost,  what  is  the  selling  price  ? 

2.  The  population  of  a  certain  village  has  increased  .25 
in  a  year.  It  numbers  now  560  inhabitants.  What  was 
the  population  one  year  ago  ? 

3.  A  person  performs  a  certain  journey  in  14.75  hours, 
traveling  3.75  miles  per  hour.  How  long  would  it  take 
him  if  he  traveled  at  the  rate  of  5.25  miles  per  hour? 

4.  Find  the  cost  of  1575  pounds  of  hay  at  $  11  per  ton 
of  2000  pounds. 

5.  What  is  the  price  per  ton  of  hay,  if  3750  pounds  are 
sold  for  $  33f ? 


24  Arithmetic 

6.  Divide  sixty  and  twelve  hundredths  times  seventy- 
two  hundredths  by  the  sum  of  thirty-two  hundredths  and 
fourteen  thousandths. 

7.  How  many  steps  of  2.5  feet  will  it  take  to  measure 
.25  mile?     (1  mi.  =  5280  ft.) 

8.  The  cost  of  .1875  of  an  article  is  equal  to  what 
decimal  of  the  cost  of  .5  of  it? 

9.  By  selling  a  house  for  $2800,  the  owner  lost  .3  of 
the  price  he  paid  for  it.     What  did  it  cost  him  ? 

10.  Find  the  value  of  £146  in  United  States  money,  £1 
being  worth  $4.8665. 

11.  Change  187.50  German  marks  to  United  States  money, 
the  mark  being  worth  23.8  cents. 

12.  At  19.3  cents  for  a  French  franc,  how  many  francs 
can  be  bought  for  $100? 

13.  Change  50   kilos  to  pounds,  the  kilo  being   2.2046 
pounds. 

14.  How  many  meters,  each  measuring  39.37  inches,  are 
there  in  a  rod  (16i  ft. )  ? 

15.  Find  the   difference  in  inches  between   a  kilometer 
(1000  meters)  and  |  mile.     (1  mi.  =  63,360  in.) 


CHAPTER   II. 
PERCENTAGE. 

39.  Preliminary  Exercises. 

Read  the  following  fractions :  y|^,  j^,  yf ^,  yf^. 

Instead  of  writing  these  fractions  with  the  denominator 
100,  we  sometimes  employ  the  sign  %,  which  is  called  per 
cent. 

40.  The  expression  7  per  cent,  therefore,  means  7  /iu7i- 
dredths.     It  is  written  7  %. 

We  can  also  w^rite  4%,  5%,  2i%,  51%,  as  decimals, 
thus  :  .04,  .05,  .0225,  .055. 

41.  Read  the  following  per  cents  as  decimals: 


1. 

2% 

6. 

m% 

11. 

6i% 

16. 

n.lo 

2. 

H% 

7. 

\1c 

12. 

3^% 

17. 

\1o 

3. 

H% 

8. 

175% 

13. 

62^% 

18. 

15i% 

4. 

10% 

9. 

87i% 

14. 

75% 

19. 

-izio 

5. 

2i% 

10. 

15i% 

15. 

37i% 

20. 

i% 

42.    Change  the  following  per  cents  to  common  fractions 
in  their  lowest  terms : 


1. 

10% 

6. 

12i% 

11. 

37^% 

16. 

6i% 

2. 

15% 

7. 

1H% 

12. 

50% 

17. 

62i  % 

3. 

25% 

8. 

35% 

13. 

150% 

18. 

i% 

4. 

75% 

9. 

45% 

14. 

661%' 

19. 

i% 

5. 

:  90% 

10. 

125% 

15. 

H% 

20. 

1% 

25 


26  Arithmetic 

43.  1.    Express  i  as  a  per  cent. 

1  _     3  0    —  fiO  0/ 

2  —  TOO  —  "^  /0« 

2.    Express  |  as  a  per  cent. 

§  =  ^^  =  37i%. 
8      100  ^ ' 

To  change  a  common  fraction  to  an  equivalent  per  cent, 
reduce  the  given  fraction  to  a  fraction  ivhose  denominator  is 
100,  and  replace  the  denominator  by  the  sign  %. 

Oral  Exercises. 

44.  Express  the  following  fractions  as  per  cents : 


1. 

1 

1 

6. 

5 
8 

9. 

3 
5 

13. 

1 

5 

17. 

7 
TO 

2. 

3 

4 

6. 

i 

10. 

4 
5 

14. 

5 
6 

18. 

9 
TO 

3. 

1 

8 

7. 

1 

5 

11. 

1 
3 

15. 

JL 
1  0 

19. 

3 

20 

4. 

3 

8 

8. 

2 
5 

12. 

2 
3 

16. 

3 

TO 

20. 

1 
40 

45.  Kead  the  following  decimals  as  per  cents: 

1.  .125 

As  a  decimal,  this  is  read  125  thousandths  ;   as  a  per  cent,  it  be- 
comes 12|  per  cent,  or  12  and  5  tenths  per  cent. 

2.  .005 

This  is  read  ^  per  cent,  or  ^  of  1  per  cent. 

3.  4.75 

This  is  read  475  per  cent. 

4.  .15  8. 

5.  .9  9. 

6.  .375  10. 

7.  2.4  11. 

46.  The  portion  of  arithmetic  that  deals  with  per  cents 

is  called  percentage. 

The  applications  of  percentage  include  no  principles  that  have  not 
already  been  studied  in  the  applications  of  fractions  and  of  decimals. 


.075 

12. 

3.125 

.0625 

13. 

.8 

.0075 

14. 

.42 

.00625 

15. 

4.2 

Percentage  27 

47.  The  three  leading  types  of  problems  in  percentage  are 
shown  in  the  following  examples  : 

1.  Find  20%  of  150. 

2.  Thirty  is  20%  of  what  number  ? 

3.  Thirty  is  what  per  cent  of  150  ? 

48.  As  examples  in  fractions,  these  have  already  been 
worked  in  the  following  form  : 

1.  Find  I  of  150. 

2.  Thirty  is  ^  of  what  number  ? 

3.  Thirty  is  what  fraction  of  150  ? 

49.  The  corresponding  decimal  examples  would  be  the 
following : 

1.  Find  .2  of  150. 

2.  Thirty  is  .2  of  what  number  ? 

3.  Thirty  is  what  decimal  of  150  ? 

50.  In  each  of  the  examples  in  percentage  (Art.  47), 
there  are  three  terms  :  the  base,  the  rate,  and  the  percentage. 
Two  of  these  are  given,  from  which  the  third  is  to  be  cal- 
culated. In  1,  the  percentage  is  required ;  in  2,  the  base ; 
in  3,  the  rate. 

TO  FIND   THE   PERCENTAGE. 

51.  Sight  Exercises. 
Find : 

1.  50  %  of  16  bushels  6.  16|  %  of  102  miles 

2.  121  %  of  24  books'^  7.  33i  %  of  84  yards 

3.  20%  of  $300  8.  10%  of  25  feet 

4.  80%  of  45  sheep  9.  66|  %  of  33  gallons 

5.  371  %  of  $  80  10.  75  %  of  $  120 


28 


Arithmetic 


Find: 

11.    25%  of  36  inches 

12. 

13. 

14. 


1%  of  $72 

150  %  of  20  inches 

1%  of  $240 


15.    1331%  of  150  pounds 

52.  Written  Exercises. 
Find  the  percentage: 


1.  52%  of  f  67.50 

In  this  example,  $67.50  is  the  base,  52  is  the 
rate  per  cent. 


16.  116|%  of  $600 

17.  125  %  of  44  feet 

18.  40%  of  $1000 

19.  87^  %  of  64  rods 

20.  31%  of  $30 


$67.50 

■  52 

13500 

3.3750 

$35.1000 

A71S.  $35.10 


To  find  the  percentage,  multiply  the  base  by  the  rate  con- 
sidered as  hundredths. 


2.   371%  of  728  acres 

371%  of  728  acres  =  728  acres  x  — ^  =  728  acres  x  5. 
^  '  100  8 


Cancel. 


Note.  In  any  example  involving  multiplication  by  a  decimal,  the 
decimal  should  be  changed  to  a  common  fraction  when  the  vs^ork  will 
be  shortened  by  such  a  change. 


3.  14%  of  375  miles 

4.  1%  of  $715 

5.  20%  of  1736  miles 

6.  25%  of  1384  acres 

7.  16|%  of  $44.64 

8.  131%  of  8021  bushels 

9.  130%,  of  88  tons 

10.  331%  of  297  cows 

11.  -1%  of  $12,000 


12.  121%  of  32.96  rods 

13.  50%  of  1984  cords 

14.  1%  of  1984  gallons 

15.  21  %  of  1260  yards 

16.  371%  of  464  books 

17.  3  %  of  1900  pupils 

18.  11  %  of  6480  pounds 

19.  J%  of  $76 

20.  75%  of  $76 


Percentage  •  29 

53.    Oral  Problems. 

1.  A  farmer  loses  3  %  of  200  tons  of  hay  by  rain.     How 
many  tons  does  he  lose  ? 

2.  There   are   36  pupils   belonging  to   a   class.      On  a 
stormy  day  25  %  are  absent.     How  many  are  present  ? 

3.  If  coffee  loses  15%  of  its  weight  in  roasting,  what  is 
the  loss  in  weight  of  a  bag  of  coft'ee  weighing  130  pounds  ? 

4.  A  man's  yearly  income  is  4%  of  $30,000.     What  is 
his  income  ? 

5.  Last  year  the  population  of  a  city  was  25,000.     What 
is  the  present  population  if  the  increase  is  4%  ? 

6.  A  farmer  put  in  his  barn  200  tons  of  hay.     What 
does  it  weigh  after  losing  2%  by  drying? 

7.  A  certain  ore  produces  8%   of  metal.     How  many 
pounds  of  metal  in  2000  pounds  of  ore  ? 

8.  A  merchant  loses  3%   on  the  cost  of  certain  goods. 
What  does  he  lose  if  the  goods  cost  $  600  ? 

9.  In  a  school  of  120  pupils  45%  are  girls.     How  many 
girls  in  the  school  ? 

10.  A  man  has  a  farm  of  200  acres,  of  which  he  has  84% 
under  cultivation.     How  many  acres  are  under  cultivation  ? 

11.  If  ^%  of  commercial  lead  consists  of  impurities,  how 
many  pounds  of  pure  metal  w^ill  there  be  in  2000  pounds  of 
the  commercial  article  ? 

12.  Of  a  flock  of  350  sheep  4%  died.  How  many 
remained  ? 

13.  A  girl  spelled  correctly  98%  of  50  words.  How  many 
did  she  miss  ? 

14.  Forty-five  per  cent  of  an  orchard  of  200  trees  are 
peach  trees.     How  many  peach  trees  are  there  ? 

15.  Mr.  Jones  sold  75%  of  his  crop  of  800  bushels  of 
corn.     How  many  bushels  did  he  keep  ? 


JO  Arithmetic 

16.  A  man  can  plow  12^%    of  a  field  in  a  day.     How 
many  days  will  it  take  him  to  plow  the  field  ? 

17.  If  sea  water  contains  2|%  of  salt,  how  many  pounds 
of  salt  can  be  obtained  from  2000  pounds  of  gea  water  ? 

18.  A  farmer  sold  all  but  25%  of  his  farm  of  480  acres. 
How  many  acres  did  he  sell  ? 

19.  A  man  who  owned  f  of  a  vessel  sold  331%   of  his 
share.     What  part  of  the  vessel  did  he  then  own  ? 

20.  An  agent  charged  me  2%  for  selling  my  house.     If 
the  house  sold  for  $  4000,  how  much  should  I  pay  the  agent  ? 

21.  If  a  broker  receives  ^%  for  buying  bonds,  how  much 
does  he  receive  on  a  purchase  of  $  8000  ? 

54.    Written  Problems. 

1.  A  man  receives  4%  yearly  on  a  capital  of  $37,500. 
What  is  his  monthly  income  therefrom  ? 

2.  The  population  of  a  certain  city  is  25,425,  of  whom 
8%  are  of  foreign  birth.     How  many  are  of  foreign  birth? 

3.  Hay,  weighing  275  tons  when  placed  in  the  barn,  has 
lost  6%  in  weight.     What  is  its  present  weight  ? 

4.  If  3|%  of  metal  can  be  obtained  from  a  certain  ore, 
how  many  pounds  of  metal  are  contained  in  2240  pounds  of 
this  ore  ? 

5.  A  merchant  lost  21%  on  goods  that  cost  him  $640. 
What  did  he  receive  for  them  ? 

6.  In  a  school  of  400  pupils,  49%  are  boys.  How  many 
girls  in  the  school  ? 

7.  The  owner  of  a  farm  of  640  acres  has  371%  under 
cultivation,  50  %  in  woods,  and  the  remainder  in  pasture. 
How  many  acres  are  in  pasture  ? 

8.  If  iron  loses  f  %  in  handling,  what  will  be  the  loss  on 
100  tons  of  2240  pounds  ? 


Percentage  3 1 

9.   Of  a  flock  of  400  sheep  4%   died,  and  25%  of  the 
remainder  were  sold.     How  many  sheep  were  left  ? 

10.  A  school  of  450  pupils  had  98%  in  attendance.  How 
many  were  present  ? 

11.  Thirty-seven  and  one  half  per  cent  of  the  trees  in  an 
orchard  of  464  trees  are  peach  trees,  25%  are  cherry  trees, 
and  the  remainder  are^apple  trees.  How  many  apple  trees 
are  there  ? 

12.  A  farmer  had  800  bushels  of  wheat,  of  which  4%  was 
lost  in  handling,  cleaning,  etc.  He  sold  25%  of  the  remain- 
der at  90^  per  bushel.  How  much  did  he  receive  for  the 
quantity  sold  ? 

13.  A  can  do  5%  of  a  piece  of  work  in  a  day,  and  B  can 
do  7^%  of  it  in  a  day.  How  many  days  will  it  take  both  to 
do  the  whole  work  ?  How  many  days  will  it  take  A  to 
do  it  alone  ?     B  to  do  it  alone  ? 

14.  If  salt  water  contains  2|%  of  salt,  how  many  pounds 
of  salt  water  will  be  required  to  produce  2000  pounds  of 
salt? 

15.  A  farmer  sold  to  one  purchaser  20%  of  his  farm  of 
480  acres,  and  to  another  25%  of  the  remainder.  How 
many  acres  had  he  remaining  ? 

16.  The  owner  of  |  of  a  vessel  sold  20%  of  his  share  for 
$  2250.     What  is  the  value  of  the  whole  vessel  at  that  rate  ? 

17.  An  agent  sold  600  bushels  of  wheat  at  90^  per  bushel. 
His  charge  was  2%  on  the  amount  received  for  the  wheat. 
How  much  did  he  return  to  the  owner  ? 

18.  A  broker  receives  \%  of  %  17,400  for  selling  bonds. 
How  much  does  he  receive  ? 

19.  A  farmer  was  offered  50^  per  hundredweight  for 
100  tons  of  hay.  He  sold  it  for  60^  per  hundredweight 
some  months  later,  after  it  had  lost  6.V%  in  weight.  Did  he 
gain  or  lose  by  refusing  the  first  offer  ? 


32  Arithmetic 

20.  How  many  pounds  of  bread  can  be  made  from  200 
pounds  of  flour,  if  the  dough  weighs  60%  more  than  the 
flour,  and  if  the  bread  weighs  15%  less  than  the  dough  ? 


TO   FIND  THE   BASE. 

55.  PreHminary  Exercises.  | 

1.  Twelve  is  J  of  what  number  ?  12  is  .25  of  what 
number?     12  is  25%  of  what  number? 

2.  Thirty-six  is  f  of  what  number  ?  36  is  .75  of  what 
number  ?     36  is  75%  of  what  number  ? 

3.  Forty  is  li  times  what  number  ?  40  is  |  of  what 
number?  40  is  1.25  times  what  number?  40  is  125%  of 
what  number? 

4.  The  percentage  40  is  1.25  times  what  base  ? 

5.  The  percentage  60  is  2  times  what  base  ?  The  per- 
centage 60  is  200%  of  what  base? 

6.  When  60  is  2  times  a  base,  how  is  the  base  found  ? 
What  sign  is  written  between  60  and  2  to  indicate  the 
operation  required  to  obtain  the  result  ? 

7.  When  12  is  i  of  a  base,  what  sign  is  written  between 
12  and  J  to  indicate  the  required  operation  ? 

8.  When  12  is  .25  of  the  base,  what  sign  do  we  write 
between  12  and  .25.  to  indicate  the  required  operation  ? 

9.  In  expressing  the  operation  of  finding  25%  of  48, 
how  is  25%  written? 

10.    How  do  we  express  the  operation  of  finding  the  base, 
when  25%  of  it  is  12? 

56.  Since  the  percentage  is  found  by  multiplying  the  base 
by  the  rate  expressed  as  hundredths. 

The  base  is  found  by  dividing  the  percentage  by  the  rate 
expressed  as  hundredths. 


Percentage  ^3 

57.  Indicating  the  number  (or  quantity)  representing  the 
base  by  B,  the  number  representing  the  rate  per  cent  by  li, 
and  the  number  (or  quantity)  representing  the  percentage 
by  P,  we  obtain  the  following : 

58.  Bx^  =  P.  B  =  P^:^. 

100  100 

* 

59.  Oral  Exercises. 

1.  21  is  50%  of  what  number? 

2.  47  is  10%  of  what  number? 

3.  18  is  25%  of  what  number? 

4.  15  is  33i  %  of  what  number  ? 

5.  27  is  9%  of  what  number? 

6.  36  is  40%  of  what  number? 

7.  42  is  116f  %  of  what  number? 

8.  55  is  125%  of  what  number? 

9.  40  is  66|%  of  what  number? 

10.  72  is  1121%  of  what  number? 

11.  Find  121%  of  80  bushels. 

12.  80  books  is  12^%  of  how  many  books  ? 

13.  What  is  20%  of  400  sheep? 

14.  $400  is  20%  of  what  sum  ? 

15.  4  miles  is  40%  of  how  many  miles  ? 

16.  What  is  1%  of  $40? 

17.  ^40isi%  of  what? 

18.  30  gallons  is  37|%  of  how  many  gallons  ? 

19.  90  rods  is  75%  of  how  many  rods  ? 

20.  36  inches  is  300%  of  how  many  inches  ? 


34  Arithmetic 

60.   Written  Exercises. 

1.    $  35.10  is  52%  of  what  sum  ? 

Base  X  .52  =  $35.10  (Art.  58). 

Base  =  $35.10  --  j%\  =  $35.10  ^  .52. 

Following    the   rule   for    division   of  ^Ql.bO    Ans. 

decimals,   the   divisor  is  changed   to   a  52)$3ol0. 

whole    number,    and    a    corresponding  ,       -        . 

change  is  made  in  the  dividend.  ^"^^ 

The    quotient,    $67.50,    is  the    base  ?^3 — 

required.  ^"" 

260 


To  find  the  base,  divide  the  percentage  by  the  rate  considered 
as  hundredths. 

2.  273  acres  is  37|^%  of  how  many  acres  ? 

Base  X  .371  —  273  acres. 

Base  =  273  acres  ^  .37^  =  273  acres  -^  f. 

To  divide  273  acres  by  f,  the  divisor  is  inverted,  according  to  the 
rule  for  division  of  fractions. 

Base  =  273  acres  x  f .     Cancel. 

3.  214  is  25%  of  what  number  ? 

Base  =  214  ~- 1 

Although  the  pupil  obtains  the  result  by  multiplying  the  percentage 
by  4,  he  has  really  divided  214  by  |.  Using  the  decimal  .25,  the  oper- 
ation would  be  indicated 

Base  =  2 14-^.25, 
while  the  pupil  would  be  expected  to  obtain  the  result  through  the 
short  method  of  multiplying  214  by  4. 

4.  Percentage  385  miles;  rate  14%.     Base? 

5.  Eate  1%  ;  percentage  ^3.75.     Base? 

6.  Base  1728  cu.  in.;  rate  37i%.     Percentage? 

7.  Rate  61%;  base  $89.28.     Percentage? 

8.  Rate  131%  ;  percentage  105  bushels.     Base? 

9.  Percentage  88  tons;  rate  16|%.     Base? 


Percentage  3  5 

10.  Base  I60  cows :  rate  3S^%.     Percentage  ? 

11.  Rate  37^%  ;  base  928  volames.     Percentage? 

12.  Percentage  24125  sq.  r(L  :  rate  12^%.    Base? 

13.  Base  8.75  gallons:  rate  160%.     Percentage? 

14.  Eate  1\%  ;  percentage  475  bosliels.     Base? 

15.  Percentage  ^175;  rate  I %.     Base? 

61.    Oral  Problems. 

1.  How  many  tons  of  hay  has  a  farmer,  if  his  loss  of 
fo  by  rain  is  12  tons  ? 

2.  On  a  stormy  day  21  pnpils  are  present,  or  80%  of  the 
class.     How  many  pnpils  belong  to  the  class  ? 

3.  A  man  gains  20%  a  year  on  the  sum  invested  in  his 
farm.     What  is  the  investment  if  his  gain  is  ^  120<3  ? 

4.  The  population  of  a  certain  city  increased  1500  in.  a 
year,  which  was  a  gain  of  3%.  Wliat  was  its  popolation  tlie 
previous  year  ? 

5.  How  many  pounds  of  ore^  yielding  6%  of  zinc,  will 
produce  120  pounds  of  zinc  ? 

6.  How  much,  did  I  borrow,  if  5%  of  the  suni  borro^eti 
is  ^20? 

7.  Of  the  pupils  of  a  certain  school  the  boys  constitube 
48%.     How  many  girls  in  the  scliooL  if  there  are  18  boys? 

8.  A  dealers  loss  of  3%  on  the  sum  paid  for  cattle 
amounts  to  ^ 60.     How  much,  did  he  pay  for  the  cattle? 

9.  How  much,  did  a  seller  receive  for  goods,  if  his  loss  on 
them  at  1%  amounted  to  ^40  ? 

10.  A  farmer  lost  20%  of  his  wheat  by  fire.    What  per 
cent  of  it  had  he  left  ? 

11.  A  farmer,  losing  20%  of  his  wheat,  had  80  bushels 
remaining.     How  manv  bushels  had  he>at  first  ? 


^6  Arithmetic 

12.  Mr.  N  lost  by  lire  40  bushels  of  a  crop  of  200  bushels 
of  wheat.     What  per  cent  of  it  did  he  lose  ? 

13.  Of  a  crop  of  300  bushels  of  wheat  a  fanner  saved  270 
bushels.     What  per  cent  did  he  lose  ? 

14.  A  man  received  $330  for  a  team  of  horses  that  cost 
him  $  300.     What  per  cent  did  he  make  on  the  cost  ? 

62.   Written  Problems. 

1.  A  dealer  loses  21%  of  a  lot  of  coal  in  handling,  screen- 
ing, etc.  How  many  tons  has  he  bought,  if  the  loss  is  203 
tons  ? 

2.  Mr.  Jones's  cotton  crop  was  77^%  of  last  year's. 
Last  year's  yield  was  60  bales  of  400  pounds  each.  How 
many  pounds  did  he  raise  this  year  ? 

3.  How  many  bushels  of  wheat  did  a  farmer  raise  this 
year,  if  last  year's  crop  of  504  bushels  was  84%  of  this 
year's  crop  ? 

4.  A  man  receives  annually,  as  rent,  10J%  of  the  cost 
of  a  house.  What  did  the  house  cost  him,  if  his  tenant 
pays  him  $  17.50  per  month  ? 

5.  After  traveling  385  miles,  a  passenger  has  completed 
14%  of  his  journey.  How  many  miles  has  he  yet  to 
travel  ? 

6.  The  Evergreen  Dairy  sold  15%  more  milk  this  year 
than  last.  The  sales  last  year  were  10,200  quarts.  How 
many  quarts  were  sold  this  year  ? 

7.  M  can  do  -j^^  of  a  piece  of  work  in  a  day.  What 
part  of  it  can  N  do  in  a  day,  if  he  can  do  20%  more  work 
than  M  ? 

8.  John  requires  30  days  to  do  a  piece  of  work  that 
James  can  do  in  25  days.  John  does  what  per  cent  less 
work  in  a  day  than  James  ? 


Percentage  37 

9.  In  washing  a  certain  quantity  of  wool  there  was  a 
loss  of  weight  of  17%.  What  did  the  unwashed  wool 
weigh,  if  the  weight  after  washing  was  332  pounds  ? 

10.  A  farmer  sold  20%  of  his  wheat  to  A,  and  20%  of 
the  remainder  to  B,  and  had  360  bushels  remaining.  How 
many  bushels  did  he  sell  to  B  ? 

TO   FIND   THE  RATE. 

63.  Preliminary  Exercises. 

1.  Twelve  is  what  fraction  of  48  ?  12  is  what  decimal 
of  48  ?     12  is  what  per  cent  of  48  ? 

2.  Thirty-six  is  wHat  fraction  of  48  ?  36  is  what  deci- 
mal of  48  ?     36  is  what  per  cent  of  48  ? 

3.  Forty  is  how  many  times  32  ?  40  is  what  fraction 
of  32  ?  40  is  how  many  hundredths  of  32  ?  40  is  what 
per  cent  of  32  ? 

4.  The  percentage  40  is  how  many  times  the  base  32  ? 

5.  The  percentage  60  is  how  many  times  the  base  30  ? 
60  is  what  per  cent  of  30  ? 

64.  Oral  Exercises. 

1.  What  per  cent  of  oO  is  40? 

2.  50  is  what  per  cent  of  40  ? 

3.  15  is  what  per  cent  of  3  ? 

4.  What  per  cent  of  15  is  3  ? 

5.  What  per  cent  of  4|-  is  1^  ? 

6.  I"  is  what  per  cent  of  f  ? 

7.  ^  is  what  per  cent  of  i  ? 

8.  What  per  cent  of  75  is  25  ? 

9.  What  per  cent  of  10  is  11  ? 

10.  120  is  90  increased  by  what  per  cent  of  90? 

11.  90  is  120  decreased  by  what  per  cent  of  120  ? 


38  Arithmetic 

12.  3  is  what  per  cent  of  4? 

13.  3  -f  1  is  what  per  cent  of  4  +  1  ? 

14.  3  —  1  is  what  per  cent  of  4  —  1  ? 

15.  Twice  3  is  what  per  cent  of  twice  4? 

65.   Written  Exercises. 

1.    $35.10  is  what  per  cent  of  $67.50? 

$67.50  X  —  =  $35.10.     (Art.  58.) 
100  ^  ^ 

—  =  $.35.10 -$67.50. 
100 


52 


Pollowing    the   rule  for  division    of    decimals,  the 

divisor  is  changed  to  a  whole  number,  and  a  corre-  $675;$  351.0 

spending  change  is  made  in  the  dividend.  ~Tr^ 

The  result  .52  gives  the  rate  expressed   as   hun-  -10  ra 
dredths.     The  answer,  therefore,  is  52  %. 

To  find  the  rate,  divide  the  percentage  by  the  base,  express- 
ing the  residt  as  hundredths. 

To  obtain  the  answer  directly  in  terms  of  the  per  cent,  we  may  pro- 
ceed as  follows : 

B   ^.$.35.10 

100      $67.50* 

^^$35.10  X  100 

~        $67.50 

Strike  out  the  decimals  in  the  numerator  and  the  denominator. 

Cancel.     The  result  gives  the  number  that  indicates  the  rate  per  cent, 

viz.,  52. 

2.    273  acres  is  what  per  cent  of  728  acres  ? 
Writing  this  in  the  form  of  an  equation,  we  have  : 

273A  =  —  of  728 A., 
100 

or  ^    X  728  =  273, 

100 

from  which  we  obtain 

27^^^^0,     Cancel. 
728 
Difficulty  in  determining  which  number  is  the  base  may  be  avoided 
by  writing  the  problem  in  the  form  of  an  equation,  as  given  above. 


Percentage  39 

3.  225  is  what  per  cent  of  15  ? 

225  =  —  X  15, 
100 

or  A  X  15  =  225, 

100 

A.  =225 
100  ~  15  ' 

B  =  ?^^.     Ans.  1500  %. 

15  ■ 

Note.  Some  pupils  think  a  result  of  1500  per  cent  looks  too  large, 
and  they  frequently  make  the  answer  15  per  cent  unless  they  are  care- 
fully drilled. 

4.  What  per  cent  of  $  750  is  ^3.75? 

—  X  750  =  3.75. 
100 

B  ^3.75 

100      750* 

„      3.75  X  100      375      ,        .        , 

-ti  = =  —  =  i.     Ans.  I  per  cent. 

750  750      ^  ^  ^ 

5.  385  miles  is  what  per  cent  of  2750  miles? 

6.  What  per  cent  of  385  miles  is  2750  miles  ? 

7.  144  en.  in.  is  what  per  cent  of  1728  cu.  in.  ? 

8.  One  fourth  of  a  clay  is  what  per  cent  of  18  hours  ? 

9.  What  per  cent  of  a  ton  is  1425  pounds  ? 

10.  108  bushels  is  what  per  cent  of  160  bushels? 

11.  $8.75  is  what  per  cent  of  $70? 

12.  $8.75  is  what  per  cent  of  $700? 

13.  $8.75  is  what  per  cent  of  $7000? 

14.  $8.75  is  what  per  cent  of  $7? 

15.  4  gallons  is  what  per  cent  of  225  gallons  ? 

16.  What  per  cent  of  165  cows  are  66  cows  ? 

17.  What  per  cent  of  |  is  f  ? 

18.  J  is  what  per  cent  of  f  ? 

19.  If  is  what  per  cent  of  2i  ? 

20.  21  is  what  per  cent  of  If? 


40  Arithmetic 

66.  Oral  Problems. 

1.  A  girl  solved  9  problems  out  of  10.  What  per  cent 
did  she  solve  ? 

2.  Out  of  200  eggs  190  were  good.  What  per  cent  of 
the  eggs  were  bad  ? 

3.  What  is  the  percentage  of  attendance,  when  24  pupils 
are  present  out  of  a  class  of  25  ? 

4.  What  per  cent  of  a  class  of  32  are  boys,  if  there  are 
20  girls  in  the  class  ? 

6.  A  man  who  borrowed  $300  pays  $18  per  year  for 
the  use  of  jt.     What  per  cent  does  he  pay  ? 

6.  A  dealer  boughlf  200  bushels  of  corn.  In  selling  it 
in  small  quantities,  he  lost  4  bushels  by  waste,  etc.  What 
per  cent  did  he  lose  ? 

7.  A  farm  200  rods  long  is  160  rods  wide.  The  width 
is  what  per  cent  of  the  length  ? 

8.  The  owner  of  5  sixths  of  a  boat  sold  1  sixth.  What 
per  cent  of  his  share  did  he  sell  ? 

9.  R  owned  |  of  a  field ;  he  sold  ^  of  his  share.  What 
per  cent  of  the  field  did  he  sell  ? 

10.  A  boy  sells  at  $  1.25  a  bushel  apples  that  cost  him  $  1 
a  bushel.     What  per  cent  of  his  receipts  does  he  gain  ? 

11.  The  population  of   a  village  increases   from  800  to 
900.     What  is  the  per  cent  of  increase  ? 

12.  If  the  population  decreases  from  900  to  800,  what  is 
the  per  cent  of  decrease  ? 

67.  Written  Problems. 

1.  A  sells  J^  of  his  land,  then  i  of  the  remainder,  then 
-1-  of  the  remainder.     What  per  cent  of  the  farm  is  unsold  ? 

2.  An  agent  receives  $350  for  selling  350  acres  of  land 
at  $50  per  acre.  What  per  cent  of  the  selling  price  does 
he  receive? 


Percentage  41 

3.  Mr.  Jones  collects  a  bill  of  ^436.50,  for  which  he 
charges  his  employer  $8.73.  What  per  cent  does  he  charge 
for  making  the  collection  ? 

4.  The  weight  of  an  ox  increased  from  1200  pounds  to 
1476  pounds.     What  per  cent  did  it  increase? 

5.  A  quantity  of  wool  weighing  380  pounds  weighed 
304  pounds  after  cleaning.     What  per  cent  did  it  lose  ? 

6.  A  baseball  club  wins  46  games  and  loses  34  games. 
What  per  cent  of  the  games  does  it  win  ? 

7.  In  making  a  trip  of  800  miles,  a  man  travels  620 
miles  by  boat  and  the  remainder  by  rail.  What  per  cent  of 
the  trip  is  made  by  rail? 

8.  The  weight  of  the  letters  on  a  certain  route  is  175 
tons  and  of  the  other  mail  matter  1125  tons.  What  per 
cent  of  the  total  weight  is  the  weight  of  each  ? 

9.  To  insure  a  house  for  $3675,  the  owner  pays  $12.25. 
What  per  cent  of  the  amount  insured  does  he  pay  ? 

10.  From  240  tons  of  sugaf  cane  there  were  extracted 
174  tons  of  juice.  What  per  cent  of  the  whole  weight 
was  the  weight  of  the  juice  ? 

11.  Of  a  farm  of  320  acres,  24  acres  are  planted  in  fruit. 
What  per  cent  of  the  farm  is  in  fruit  ? 

12.  By  draining  a  certain  field,  a  farmer  increases  the 
yield  of  hay  from  37^  tons  to  50  tons.  What  per  cent  is  the 
increase  ? 

13.  Two  different  fertilizers  are  applied  to  the  same 
quantity  of  land.  On. one  the  yield  is  320  bushels  of  oats, 
weighing  30  pounds  to  the  bushel ;  on  the  other  the  yield  is 
400  bushels,  weighing  32  pounds  to  the  bushel.  By  what  per 
cent  does  the  weight  of  the  latter  crop  exceed  that  of  the 
former  ? 

14.  A  piece  of  beef  weighing  18J  pounds  before  cooking 
weighs  15  pounds  when  roasted.  What  per  cent  of  its 
weight  is  lost  ? 


42  Arithmetic 

15.  How  many  pounds  of  bread  can  be  made  from  a 
bushel  of  rye  weighing  56  pounds,  if  the  flour  weighs 
25%  less  than  the  grain  and  the  bread  weighs  33^-%  more 
than  the  flour  ? 


AMOUNT  AND  DIFFERENCE. 

68.  PreHminary  Exercises. 

1.  Multiply  15  by  l^.     Increase  15  by  i  of  itself.     Find 
f  of  15. 

2.  Diminish  15  by  i  of  itself.     Find  4  of  15. 

3.  If   f   of   a   number    is    18,    what   is    the    number  ? 
What  number  increased  by  i  of  itself  gives  18  as  the  result  ? 

4.  What   number  diminished  by  ^  of  itself  gives  a  re- 
mainder of  12  ? 

5.  Fifteen   increased  by  20%   of    itself  equals  what? 
18  equals  what  number  increased  by  20%  of  itself? 

69.  Amount  =  Base  +  Percentage. 
Difference  =  Base  —  Percentage. 

70.  Sight  Exercises. 

1.  Increase  18  by  50%  of  itself.     Amount? 

2.  Diminish  150  by  33^%  of  itself.     Difference  ? 

3.  Amount  27;  rate  50%.     Base? 

4.  Difference  100;  rate  331%.     Base? 

5.  What  number  increased  by  25%  of  itself  becomes  45? 

6.  What  number  decreased  by  50%  of  itself  becomes  18  ? 

7.  A  number  increased  by  12^%  of  itself  becomes  108. 
What  is  the  number  ? 

8.  Difference  45;  rate  16|%.     Base? 

9.  What  number  increased  by  20%  of  itself  becomes 
420? 


Amount  and  Difference  43 

10.  Difference  28;  rate  121%.     Base? 

11.  A  number  increased  by  75%  of  itself  becomes  140. 
Find  the  number. 

12.  What  number  diminished  by  2o(^o  of  itself  becomes 
150? 

13.  Amount  28;  rate  16J%.     Base? 

14.  A  number  diminished  by  33|%  of  itself  becomes 
60.     Find  the  number. 

15.  Rate  66|%;  amount  60.     Base? 

16.  What  number  increased  by  25%  of  itself  becomes  75  ? 

17.  What  number  diminished  by  75%  of  itself  becomes 
120? 

18.  Rate  871%  ;  difference  300.     Base? 

19.  AVhat  number  increased  by  50%  of  itself  becomes 
300? 

20.  Difference  140;  rate  121%.     Base? 

71.  Written  Problems. 

1.  What  number  increased  by  27%  of  itself  equals  508  ? 

In  this  example  the  amount,  508,  is  given  and  the  rate,  27%,  from 
which  the  base  is  to  be  obtained.  The  amount,  508,  equals  the  num- 
ber +  27%  of  the  number,  or  127%  of  the  number,  which  is  1.27  times 

the  number.  ,^„ 

400 

If  1.27  times  the  number  =  508,  ,^„Trrrr:r 

*!,  u  cno         -.    o-r  127)00800 

the  number  =  508  -r-  1.27. 

508 

Make  the  divisor  a  whole  number,  and  make  a  cor-  ' 

responding  change  in  the  dividend.  Ans.  400. 

To  find  the  base  when  the  amount  and  the  rate  are  given, 
divide  the  amount  by  1  added  to  the  rate  considered  as 
hundredths. 

2.  What  number  diminished  by  19  %  of  itself  equals  324  ? 
In  this  example  324  represents  the  difference.   Number  =  324  -f-  .81. 


44  Arithmetic 

To  find  the  base  when  the  difference  and  the  rate  are  given, 
divide  the  difference  by  1  diminished  by  the  rate  considered  as 
hundredths. 

3.  A  certain  number  increased  by  33^  %  of  itself  equals 
576.     Find  the  number. 

576  -  1.331  =  576  --  |. 

4.  After  losing  15%  of  its  weight  in  roasting,  a  bag  of 
coffee  weighs  110|^  pounds.  What  was  its  weight  before 
roasting  ? 

5.  A  city  has  increased  in  a  year  15%  in  population. 
The  present  population  is  32,430.  What  was  the  population 
the  year  before  ? 

6.  A  drover  sold  cattle  at  an  average  of  $  58.80,  which 
was  an  increase  of  40%  over  the  price  he  received  for  an- 
other lot.     What  did  he  receive  per  head  for  the  latter  ? 

7.  In  a  year  the  number  of  manufacturing  establish- 
ments has  increased  4%,  the  present  number  being  215,878. 
How  many  were  tli^re  the  year  before  ? 

8.  In  a  year  the  employees  increased  16%  in  number, 
there  being  now  5,469,429.  What  was  their  number  the 
previous  year  ? 

9.  A  farm  was  sold  for  $11,340,  an  advance  of  40%  of 
its  cost.  How  many  dollars  more  than  its  cost  did  the 
seller  receive  ? 

10.  In  a  certain  school  the  number  of  girls  is  10%  less 
than  the  number  of  boys.  There  are  279  girls.  How  many 
pupils  are  there  in  the  school  ?       » 

11.  A  piece  of  property  was  sold  for  $8000,  which  was 
150%  in  excess  of  its  cost.     What  was  the  cost? 

12.  The  yield  of  an  orchard  was  405  bushels,  which  was 
19%  less  than  the  yield  of  the  preceding  year.  How  many 
bushels  did  it  yield  the  year  before? 


Amount  and   Difference  45 

13.  What  number  decreased  by  26%  of  itself  becomes 
262.32  ? 

14.  A  man  loses  17|^%  of  the  cost  of  a  piece  of  cloth  by 
selling  it  for  $  122.10.     AYhat  was  the  cost  ? 

15.  After  selling  68%  of  his  sheep,  a  farmer  still  had  80. 
How  many  had  he  originally  ? 

16.  I  sold  a  horse  for  $  180,  which  was  20%  less  than  its 
cost.     What  did  I  lose  by  the  sale  ? 

17.  I  sold  a  horse  for  $180,  which  was  20%  more  than 
its  cost.     What  was  my  profit  ? 

18.  What  was  a  dealer's  net  gain  or  loss  on  two  horses 
sold  at  $  150  each,  on  one  of  which  he  gained  25%,  and  on 
the  other  he  lost  25%? 

19.  After  selling  77%  of  his  crop  of  potatoes,  a  farmer 
has  529  bushels  left.     How  many  bushels  in  the  crop  ? 

20.  A  man  traveled  62|%  of  his  journey  by  train,  and 
the  remainder,  243  miles,  by  steamer.  How  many  miles  did 
he  travel  ? 

72.   Oral  Problems. 

1.  A  city  increased  10%  in  population  in  a  year.  It 
then  had  22,000  inhabitants.  How  many  had  it  the  pre- 
vious year  ? 

2.  A  dealer  gained  25%  by  selling  a  book  for  $1.50. 
What  did  it  cost  ? 

3.  After  selling  20%  of  his  chickens,  John  has  20  left. 
How  many  had  he  at  first  ? 

4.  The  minuend  is  50.  The  remainder  is  25%  of  the 
subtrahend.     What  is  the  remainder  ? 

5.  What  number  increased  by  75%  of  itself  equals  12^? 

6.  William  has  20  apples  ;  he  has  25%  more  apples  than 
John.     How  many  has  John  ? 


46  Arithmetic 

7.  John  has  16  apples;  he  has  20%  fewer  apples  than 
William.     How  many  has  William  ? 

8.  A  lot  was  sold  for  $600,  which  was  200%  more  than 
it  cost.     What  did  it  cost  ? 

9.  Twenty  per  cent  of  a  farmer's  cows  are  Jerseys ;  the 
remaining  24  are  Ayrshires.     How  many  cows  has  he  ? 

10.  There  are  2  more  boys  than  girls  in  a  class  in  which 
45%  of  the  pupils  are  girls.  How  many  pupils  are  there 
in  the  class  ? 

73.   Miscellaneous  "Written  Drills. 

1.  Eate  11%  ;  percentage  1001  men.     Base  ? 

2.  Amount  $  287.50  ;  base  $  250.     Rate  ? 

3.  Eate  13%  ;  amount  1469  acres.    Percentage? 

4.  Base  350  bushels  ;  rate  21%.     Amount? 

5.  Bate  16%  ;  base  325  pounds.     Difference? 

6.  Amount  $262.60;  rate  102%.     Base? 

7.  Base  416  cows  ;  percentage  728  cows.     Bate  ? 

8.  Difference  456  tons  ;   rate  24%.     Percentage  ? 

9.  Percentage  147  sheep  ;  rate  87|^%.     Amount  ? 

10.  Base  125  bales;  rate  48%.     Difference? 

11.  Bate  23%  ;  difference  201  gallons.     Base  ? 

12.  Percentage  43  miles  ;  amount  243  miles.    Rate  ? 

13.  Rate  160%  ;  base  225  trees.     Percentage  ? 

14.  Base  $250;  rate  350%.     Amount? 

15.  Percentage  117  horses  ;  rate  39%.     Difference? 

16.  Rate  11%;  amount  $304.50.     Base? 

17.  Amount  250  rods ;  base  210  rods.     Rate  ? 

18.  Difference  297f  oz. ;  rate  |%.     Percentage  ?  . 

19.  Rate  |^%  ;  percentage  114|^.    Amount  ? 

20.  Base  264  quarts 5  rate  75%.     Difference? 


Profit  and   Loss  47 

PROFIT   AND   LOSS. 

74.  The  profit  or  the  loss  in  a  business  transaction  is  the 
difference  between  the  cost  and  the  selling  price. 

When  the  expense  of  conducting  business  or  of  selling 
goods  is  deducted  from  the  profit  or  added  to  the  loss,  the 
result  is  the  net  gain  or  loss. 

75.  In  problems  in  profit  and  loss  the  cost  is  the  base, 
the  profit  or  the  loss  is  the  percentage,  and  the  part  of  the 
cost  represented  by  the  profit  or  the  loss  expressed  in  hun- 
dredths is  the  rate. 

76.  When  the  transaction  produces  a  profit,  the  selling 
price  is  the  amount  ;  when  it  causes  a  loss,  the  selling  price 
is  the  difference. 

77.  Oral  Exercises. 

1.  Cost  $  24  ;  gain  25%.     Selling  price  ? 
Suggestion.     Use  a  short  method  wherever  practicable. 

2.  Cost  $  1.50;  selling  price  $  2.     Gain  %  ? 

3.  Selling  price  $  240 ;  loss  20%.     Cost  ? 

4.  Selling  price  $240;  gain  20%.     Cost? 

5.  Cost  $30;  loss  $  6.     Loss%  ? 

6.  Selling  price  $  24  ;  loss  $  6.     Loss  %  ? 

7.  Gain  $50;  selling  price  $75.     Gain  %  ? 

8.  Selling  price  $75;  gain  66|%.     Cost? 

9.  Selling  price  $  75  ;  loss  66|%,.     Cost  ? 
10.  Cost  5^;  selling  price  16^.     Gain  %  ? 

78.  Written  Exercises. 

1.  Cost  $250;  gain  28%.     Selling  price  ? 

2.  Cost  $1.50;  selling  price  $  1.62i      Gain  %  ? 


48  •  Arithmetic 

3.  Selling  price  $182;  gain  16|%.     Cost? 

4.  Selling  price  $67.20;  loss  16%.     Cost? 

5.  Cost  $  375.96  ;  loss  $  62.66.     Loss  %  ? 

6.  Selling  price  $  69 ;  loss  $  6.     Loss  %  ? 

7.  Gain  $  9.76  ;  selling  price  f  133.01.     Gain  %  ? 

8.  Selling  price  $  93.60 ;  gain  8i%.     Cost  ? 

9.  Selling  price  $  1274.10  ;  loss  7%.     Cost  ? 
10.  Cost  $  864  ;  selling  price  $936.     Gain  %  ? 

79.    Oral  Problems. 

1.  A  house  that  cost  $5000  was  sold  for  $6000.     What 
was  the  gain  per  cent? 

2.  A  farm  costing   $5000  was  sold  at  a  gain  of  25%. 
What  was  the  gain  ? 

3.  If  I  buy  eggs  at  30  cents  a  dozen  and  sell  at  a  loss 
of  10%,  at  what  price  do  I  sell  them? 

4.  Berries  are  sold  at  24  cents,  which  is  20%  more  than 
they  cost.     How  much  did  they  cost? 

5.  What  is  the  cost  of  berries  sold  for  24  cents  at  a  loss 
of  20%? 

6.  What  per  cent  will  be  gained  by  buying  beans  at  5 
cents  a  quart  and  selling  them  at  7  cents  ? 

7.  Cloth  costing    $1.60  a  yard   is    sold    at  a    gain   of 
121%.     What  is  the  selling  price? 

8.  What  per  cent  is  lost,  when  tea  costing  40  cents  per 
pound  is  sold  for  24  cents  per  pound  ? 

9.  By  selling  an  article  for  $30,  I   lost  25%.     What 
did  it  cost  ? 

10.    I  gained  $500  on  a  house,  which  was  33^%  of  the 
cost.     How  much  did  it  cost  ? 


Profit  and   Loss  49 

80.    Written  Problems. 

1.  Wheat  bought  at  84^  a  bushel  is  sold  at  98^.  What 
is  the  gain  per  cent  ? 

2.  A  man  bought  75  barrels  of  apples  at  $2.60  per 
barrel  and  sold  them  at  a  gain  of  15%.  What  was  the 
selling  price  per  barrel?     What  was  his  total  profit? 

3.  A  merchant  sold  cloth  at  $1,871  a  yard,  which  was 
25%  more  than  it  cost.     What  did  it  cost  ? 

4.  Mr.  Rafferty  sold  a  house  for  $  2800,  thereby  losing 
20%.     At  what  price  would  he  have  to  sell  it  to  gain  20%  ? 

5.  A  watch  is  sold  for  $90,  which  is  145%  more  than 
it  cost.     What  is  the  gain  ? 

6.  Berries  bought  at  $4.80  a  bushel  are  sold  at  18^ 
a  quart.     Find  the  per  cent  of  gain  or  of  loss. 

7.  A  horse  was  sold  for  $20  less  than  it  cost,  which  was 
a  loss  of  13i%.     Find  the  selling  price. 

8.  By  selling  an  article  for  $  (y.56  more  than  he  paid  for 
it,  a  merchant  gained  16%.     Find  the  cost. 

9.  I  sold  a  horse  for  $175,  which  was  12^%  less  than  I 
paid  for  it.  How  much  would  I  have  gained,  if  I  had  sold 
it  for  $227? 

10.  Cloth  which  cost  $1.26  a  yard  is  marked  o3J%  above 
cost.  What  is  the  per  cent  gain,  if  it  is  sold  at  10%  below 
the  marked  price? 

11.  A  speculator  bought  5000  bushels  of  corn  at  49^  a 
bushel.  He  sold  40%  of  it  at  an  advance  of  16%,  30%  of 
it  at  a  loss  of  10%,  and  the  rest  at  50  j^  a  busheh  Find  out 
how  much  he  gained. 

12.  A  newsdealer  bought  400  papers  at  $  1.50  per  hun- 
dred and  sold  376  of  them  for  2^  each,  the  remainder  being 
destroyed.     What  was  his  gain  per  cent  ? 

13.  A  farm  of  430  acres  costing  $20j640  was  sold  at  a 
profit  of  15%.     At  what  price  per  acre  was  it  sold? 


50  Arithmetic 

14.  How  much  did  I  lose  on  goods  for  which  I  paid 
1836,  my  loss  being  61%  ? 

15.  A  merchant  sold  goods  to  the  amount  of  $3650.40, 
on  which  his  profit  was  17%.     What  was  the  cost  ? 

16.  A  farmer  sold  a  piece  of  land  for  $225  more  than 
its  cost,  gaining  7^%.     What  did  he  receive  for  the  land? 

17.  A  dealer  buys  1875  bushels  of  wheat  at  90  cents  a 
bushel.  He  sells  it  at  $1  a  bushel,  after  the  quantity 
is  diminished  4  %  by  drying  out,  etc.  What  per  cent  does 
he  gain  ? 

18.  A  grocer  buys  400  pounds  of  tea  at  50^  per  pound. 
He  sells  J  of  it  at  a  profit  of  5^  per  pound,  and  the  re- 
mainder at  a  profit  of  10  ^  per  pound.  W^hat  is  his  gain  per 
cent  on  the  whole  ? 

19.  A  merchant  bought  400  yards  of  cloth  at  $1.25  per 
yard.  He  sells  f  of  it  at  $  1.30  per  yard.  At  what  rate 
must  he  sell  the  remainder  to  realize  a  profit  of  15%  on 
the  entire  quantity  ? 

20.  Paid  141^^  per  pound  for  a  130  pound  bag  of  coffee 
and  1^  per  pound  for  roasting.  The  loss  in  weight  is  20%, 
and  the  roasted  coffee  is  sold  at  21  ^  per  pound.  What  is 
the  gain  per  cent  on  the  total  cost  of  the  roasted  coffee  ? 

21.  A  farmer  buys  75  sheep,  i  of  them  at  $4.20  each, 
^  of  them  at  $3.80  each,  and  the  remainder  at  $3  each. 
After  spending  $  121  for  food,  care,  etc.,  he  sells  them  at  an 
average  of  $6  each.  What  per  cent  does  he  gain  on  his 
total  outlay  ? 

22.  The  owner  of  a  farm  of  160  acres,  which  cost  him 
$40  per  acre,  sells  ^  of  it  at  $30  per  acre  and  the  remainder 
at  $50  per  acre.     What  per  cent  does  he  gain  ? 

23.  By  the  sale  of  a  farm  at  an  advance  of  $  425  over  the 
cost,  the  seller  makes  a  profit  of  8^%.  How  many  acres 
does  the  farm  contain,  if  the  selling  price  is  $43.40  per 
acre? 


Commission   and   Brokerage  51 

24,  A  man  bought  160  measured  bushels  of  oats,  weigh- 
ing 30  pounds  to  the  bushel,  for  §80.  He  sold  them  by 
weight  at  06  cents  per  bushel  of  32  pounds.  What  per  cent 
did  he  gain  or  lose  ? 

COMMISSION   AND   BROKERAGE. 

81.  When  the  owner  of  property  desires  to  sell,  he 
frequently  has  recourse  to  a  real  estate  agent.  If  the 
agent  secures  a  purchaser,  he  charges  the  seller  a  certain 
per  cent  of  the  selling  price  as  his  commission. 

82.  The  farmer  ships  his  butter,  eggs,  berries,  etc.,  to  a 
commission  merchant  to  be  sold.  The  latter  remits  to  the 
farmer  the  amount  obtained,  after  deducting  a  commission 
on  this  amount,  and  also  any  charge  that  he  may  have  paid 
for  freight  or  other  expense. 

83.  Large  business  houses  employ  persons  to  buy  or  to 
sell  goods  for  them.  These  agents,  factors,  or  salesmen  are 
usually  paid,  in  whole  or  in  part,  by  a  certain  percentage  of 
the  value  of  the  goods  they  buy  or  sell,  as  a  commission,  or 
brokerage. 

84.  A  commission  merchant  is  usually  a  person  who  actu- 
ally handles  the  goods  consigned  to  him.  He  pays  the 
charges  for  freight,  carts  the  goods  to  his  warehouse  or 
salesroom,  and  sells  them  at  once  at  the  market  rate,  unless 
he  is  instructed  by  the  consignor  to  hold  them  for  a  better 
price. 

85.  The  term  broker  is  generally  applied  to  an  agent 
who  buys  or  sells  any  kind  of  merchandise  without  actually 
receiving  or  handling  the  commodities.  He  may  buy  for  a 
New  York  shipper  100,000  bushels  of  wheat  stored  in  a 
Chicago  warehouse  or  on  board  of  a  tram  on  its  way  to  the 
coast.     He  merely  fixes  the  price  and  arranges  for  the  deliv- 


£2  Arithmetic 

ery  to  the  purchaser.      His  brokerage  may  be  a  fixed  sum 
per  bushel  or  a  percentage  of  the  price,  as  may  be  agreed. 

86.  A  person  is  frequently  employed  at  a  commission  to 
collect  money  due. 

87.  Commission,  or  brokerage,  is  the  sum  paid  to  any  per- 
son who  acts  for  another  as  agent,  commission  merchant,  broker, 
or  the  like. 

88.  The  expression  net  proceeds  indicates  the  sum  remitted 
to  the  owner  of  goods  after  all  expenses  attending  the  sale 
are  deducted,  including  the  commission,  or  brokerage. 

89.  In  commission,  or  brokerage,  the  base  is  the  value  of 
goods  bought  or  sold,  the  amount  collected,  etc.  The  per- 
centage is  the  sum  paid  the  agent  for  his  services.  The 
difference,  or  proceeds,  is  the  sum  received  for  goods  sold 
less  the  commission.  The  amount  is  the  cost  of  goods 
bought  plus  the  commission  for  buying. 

90.  Oral  Problems. 

1.  An  agent  collected  a  debt  of  $400.  What  was  his 
commission  at  5%? 

2.  At  2%,  what  would  a  commission  merchant  receive 
for  selling  $  120  worth  of  potatoes  ? 

3.  A  boy  selling  eggs  at  3%  commission  made  $1.50. 
How  many  eggs  at  25  cents  a  dozen  did  he  sell  ? 

4.  How  much  would  a  broker  receive,  at  2^%  ,  for  buy- 
ing cotton  to  the  value  of  f  16,000  ? 

5.  A  man  bought  through  an  agent  a  farm  costing 
$  10,000.  What  was  the  sum  paid,  including  the  commis- 
sion at  1%  ? 

6.  What  sum  was  received  for  goods  on  which  the  com- 
mission at  4%  was  $  60  ? 


Commission  and   Brokerage  ^2 

7.  The  total  cost  of  an  article  bought  through  a  broker 
was  $  210,  including  his  commission  at  5%.  What  did  the 
broker  pay  for  it  ? 

8.  How  much  would  a  real  estate  agent  receive  for  sell- 
ing a  house  for  $3000  at  a  commission  of  2%  ? 

9.  A  collector  received  $  25  for  collecting  a  debt.  His 
commission  was  2|-%.     What  sum  did  he  collect? 

10.  A  commission  merchant  remits  to  a  farmer  $  97 
after  deducting  his  commission  at  3%.  For  how  much 
were  the  goods  sold  ? 

11.  An  auctioneer  sold  goods  amounting  to  $  1000. 
What  were  the  net  proceeds  after  the  commission  of  3% 
and  )p20  for  other  expenses  were  deducted? 

12.  How  much  wheat  can  be  bought  through  a  broker 
for  $950,  when  wheat  sells  for  94|-  cents  per  bushel,  and 
the  broker  receives  ^  cent  brokerage  per  bushel  ? 

13.  What  would  be  the  commission  at  2|%  on  a  bale  of 
cotton  weighing  480  pounds,  sold  for  10  cents  a  pound  ? 

14.  A  lawyer  collected  GC)i%  of  a  debt  of  $600.  What 
did  he  remit  to  his  client  after  deducting  5%  commission? 

15.  A  commission  merchant  charged  a  customer  $2100. 
This  included  the  sum  paid  for  the  goods  and  a  commission 
of  5%  on  the  purchase  price.  What  was  the  cost  of  the 
goods  ? 

91.    "Written  Problems. 

1.  A  broker  buys  225  bales  of  cotton,  averaging  480 
pounds  per  bale,  at  10.45  cents  per  pound.  What  is  his 
brokerage  at  2^%  ? 

$.1045  X  480  X  225  x^\. 

2.  At  4%  brokerage,  what  will  a  broker  receive  for  sell- 
ing 250  bags  of  coffee,  weighing  130  pounds  each,  at 
9.55  cents  per  pound  ? 


54  Arithmetic 

3.  A  commission  merchant  sold  80  barrels  of  apples  at 
$2.75  per  barrel.  How  much  did  he  remit  to  the  owner 
after  deducting  his  commission  of  2|  %  ? 

4.  An  agent's  commission  at  2|%  was  $  550  on  the  sale 
of  400  acres.     At  what  price  did  the  land  sell  per  acre  ? 

5.  A  collector  charging  2|-%  commission  received  $  21.50 
for  collecting  43%  of  a  debt.  What  sum  was  collected,  and 
how  much  does  the  debtor  still  owe  ? 

6.  A  merchant  buys  through  an  agent  130  yards  of  silk 
at  $1,621  a  yard.  If  he  pays  2^%  brokerage  and  other 
expenses  amount  to  $1.75,  what  is  the  lowest  price  per  yard 
at  which  the  silk  can  be  sold  without  loss  ? 

7.  Mr.  Mills  bought  goods  through  an  agent  at  a  com- 
mission of  2%.  The  cost  of  the  goods  added  to  the  commis- 
sion amounted  to  $  994.50.     What  was  the  commission  ? 

8.  A  broker  buys  for  me  100  barrels  of  beef  at  $  18.25 
per  barrel.  How  much  per  barrel  does  the  beef  cost  me, 
including  the  commission  at  3%  and  other  charges  amount- 
ing to  $15.25? 

9.  How  much  do  I  save  by  paying  a  broker  i%  for  buy- 
ing 400  bushels  of  wheat  at  95  cents  a  bushel,  instead  of 
giving  him  a  commission  of  i  cent  per  bushel  ? 

10.  A  commission  merchant  sold  for  a  farmer  560  baskets 
of  peaches.  The  net  proceeds  were  $470,  after  deducting 
5%  commission  and  $8.80  expenses.  What  did  the  peaches 
bring  per  basket  ? 

11.  A  broker  bought  for  a  produce  dealer  560  baskets  of 
peaches  at  90  cents  a  basket.  How  much  per  basket  did 
they  cost  the  dealer,  if  he  paid  5%  brokerage  and  $8.80  for 
freight,  etc.  ? 

12.  A  broker  was  paid  $9.75  for  buying  grain  of  the 
value  of  $  7790.25  at  99|^  cents  per  bushel.  What  was  his 
brokerage  per  bushel  ? 


Insurance  ^^ 

13.  What  are  an  auctioneer's  fees  for  selling  goods  to  the 
amount  of  $7500  at  5%  on  the  first  thousand  dollars,  2i% 
on  the  next  four  thousand  dollars,  and  1%  on  all  above  five 
thousand  dollars  ? 

14.  A  collector  remitted  $  93.60  after  deducting  21%  of 
the  sum  collected.     How  much  did  he  collect  ? 

15.  The  net  proceeds  of  a  sale  of  property,  after  the  de- 
duction of  the  agent's  commission  of  14%  and  $29.52  for 
advertising  and  other  expenses,  were  $  8200.  What  did  the 
property  bring  at  the  sale? 

INSURANCE. 

92.  Insurance  is  a  contract  to  pay  to  a  person,  or  his  repre- 
sentatives, a  sum  of  money  in  the  event  of  a  loss. 

The  two  chief  classes  of  insurance  are  property  insurance 
and  personal  insurance. 

93.  There  are  various  forms  of  property  insurance. 
Fire  insurance,  the  most  usual  form,  is  an  agreement  to 
indemnify  the  owner  of  a  building  or  other  property  lost 
by  fire.  Marine  insurance  covers  loss  by  fire,  shipwreck, 
and  acts  of  piracy.  Insurance  against  losses  by  theft, 
cyclone,  lightning,  and  breakage  of  plate  glass  windows, 
may  also  be  obtained. 

94.  Life  insurance  is  the  chief  form  of  personal  insur- 
ance. The  essential  feature  is  an  agreement  to  pay  some 
person  a  certain  sum  at  the  death  of  the  person  insured. 
Endoimnent  insurance  is  a  combination  of  investment  and 
simple  life  insurance,  in  which  the  insurer  agrees  to  pay  a 
certain  sum  at  a  time  named  or  at  the  death  of  the  person 
insured  if  he  dies  before  that  time.  Accident  insurance  is 
an  agreement  to  pay  a  fixed  sum  monthly  or  weekly  in  case 
the  insured  person  is  injured  through  accident,  and  a  fixed 
sum  in  case  death  results  from  an  accident. 


^6  '  Arithmetic 

95.  The  business  of  insurance  is  conducted  by  incor- 
porated companies.  The  company  issues  to  the  person 
insured  a  written  contract,  termed  a  policy  of  insurance. 
This  specifies  the  amount  for  which  the  property  or  life  is 
insured,  called  the  face  of  the  policy,  the  sum  to  be  paid 
to  the  company,  called  the  premium,  and  the  other  details 
of  the  agreement. 

96.  In  fire  insurance  problems  the  face  of  the  policy  is 
the  base,  the  premium  is  t\vQ  percentage. 

Note.  An  insurance  policy,  which  may  be  obtained  from  a  local 
agent,  should  be  brought  to  the  classroom  and  its  provisions  read  and 
discussed. 

97.  Oral  Problems. 

1.  What  is  the  cost  of  insuring  furniture  for  $  750  at 
2%  ? 

2.  The  owner  insures  his  property  at  f  %,  paying  $15 
premium.     What  is  the  face  of  his  policy  ? 

3.  A  merchant  insures  his  goods  for  $  12,000  at  90  cents 
per  $100.     What  is  his  premium? 

4.  What  rate  per  cent  corresponds  to  60  cents  on  $100? 

5.  A  house  costing  $12,000  is  insured  for  f  of  its  value 
at  1  %.     What  is  the  premium  ? 

6.  I  paid  $  15  for  insuring  my  house  at  i%.  What  was 
the  face  of  my  policy  ? 

7.  A  premium  of  $60  at  the  rate  of  li%  is  paid  to  in- 
sure a  store  for  |  of  its  value.  What  is  the  value  of  the 
store  ? 

8.  If  a  company  insures  property  to  the  amount  of 
$800,000  at  an  average  of  1%,  and  pays  losses  and  expenses 
amounting  to  J%  of  the  total  amount  insured,  how  much 
does  it  gain  ? 


Insurance  57 

9.  My  property  is  insured  through  an  agent  for  $  10,000 
at  j%.  The  agent  is  paid  by  the  company  20%  of  the 
premium.     How  much  does  the  agent  receive  ? 

10.  Mr.  Taylor  has  insured  his  property  for  20  years 
for  80%  of  its  value,  paying  an  annual  premium  of  1%. 
If  the  property  is  worth  $10,000,  what  are  his  total 
payments  ? 

98.    Written  Problems. 

1.  Find  the  premium  on  a  policy  for  $8500  at  2^%. 

2.  A  company  insures  a  house  for  3  years  upon  an 
advance  payment  of  2  years'  premiums.  What  advance 
payment  will  insure  a  house  worth  $5000  for  80%  of  its 
value  for  3  years,  the  rate  for  one  year  being  f  %  ? 

3.  What  will  be  the  total  cost  of  insuring  a  factory  for 
$100,000  and  the  contents  for  $35,000  at  60^  per  $100  on 
the  former,  and  90^  per  $  100  on  the  latter  ? 

4.  A  shipment  of  5000  bushels  of  corn  worth  60  cents 
per  bushel  is  insured  for  |  of  its  value  at  1|%.  What  is 
the  premium  ? 

5.  What  commission  does  an  agent  receive  for  insuring 
a  factory  worth  $75,000  for  |  of  its  value  at  1^%,  if  the 
company  allows  him  20%  of  the  premium  ? 

6.  I  obtain  a  policy  for  3  years  by  the  payment  of 
double  the  yearly  rate.  If  the  latter  is  1|^%,  what  does 
my  insurance  cost  me  annually  on  property  insured  for 
$11,250? 

7.  A  building  was  insured  for  |  of  its  value^at  2%,  the 
premium  being  $  200.    W^hat  was  the  value  of  the  building  ? 

8.  A  storehouse  containing  8000  bushels  of  wheat  worth 
90  cents  a  bushel  is  totally  destroyed  by  fire.  If  the 
owner  of  the  wheat  is  insured  for  87^%  of  its  value  at  f%, 
what  is  his  loss,  including  the  premium? 


58  Arithmetic 

9.  A  house  worth  $5000  is  insured  for  80%  of  its  value 
at  li%.  If  the  company  pays  a  loss  of  $3000  after  it  has 
received  24  annual  premiums,  how  much  does  it  pay  in 
excess  of  the  premiums  ? 

10.  The  owner  of  a  barn  pays  $90  premium  on  the  con- 
tents at  21%,  and  $27  on  the  building  at  f%.  What  is 
the  total  value  of  the  property,  if  the  contents  are  insured 
for  1  of  their  value  and  the  building  is  insured  for  |  of  its 

value  ? 

TAXES   AND   DUTIES. 

99.  The  nation,  the  state,  the  county,  the  town,  the  city, 
the  village,  the  school  district  —  all  require  money  for  public 
uses.  The  expenses  of  the  national  government  amount  to 
over  a  million  dollars  a  day  :  public  improvements,  public 
institutions,  and  salaries  of  officials  require  immense  sums 
of  money. 

100.  Money  for  public  purposes  is  raised  chiefly  in  two 
ways : 

1.  By  a  tax  on  property. 

2.  By  a  tax  on  goods  manufactured  at  home  and  on  goods 
brought  into  this  country. 

TAXES. 

101.  The  expenses  of  states,  counties,  and  all  minor  sub- 
divisions are  met  chiefly  by  a  tax  on  property.  In  some 
communities  a  poll  tax  also  is  levied,  which  is  a  tax  usually 
of  one  or  two  dollars  payable  by  all  male  citizens  over 
twenty-one  years  of  age. 

102.  Property  is  of  two  kinds  : 

1.  Real  estate,  or  fixed  property ;  such  as  land,  buildings, 
mines,  railroads,  etc. 

2.  Personal,  or  movable,  property ;  such  as  tools,  ma- 
chinery, furniture,  jewelry,  etc. 


Taxes  and   Duties  59 

103.  For  purposes  of  taxation  the  value  of  each  person's 
property  is  determined  by  men  called  assessors,  who  are 
selected  for  this  purpose.  The  value  fixed  by  the  assessors 
is  called  the  assessed  value  and  is  generally  less  than  the 
actual  market  value. 

104.  The  tax  rate  is  then  determined  by  dividing  the 
amount  to  be  raised  by  taxation  by  the  total  assessed  value. 
The  rate  may  be  expressed  in  various  ways ;  it  may  be 
given  as  a  per  cent,  as  so  many  mills  or  cents  on  the  dollar, 
as  so  many  dollars  on  the  thousand,  etc. 

105.  Taxes  are  generally  paid  to  a  special  officer  called 
a  collector  or  receiver  of  taxes,  who  receives  either  a  fixed 
salary  or  a  commission  on  the  sum  collected. 

106.  The  base  in  tax  problems  is  the  assessed  value,  the 
tax  being  the  percentage. 

107.  Oral  Exercises. 
Find  the  tax : 

1.  On  $1000  at  4  mills  on  the  dollar. 

2.  On  $3000  at  ^fc. 

3.  On  $12,000  at  $15  on  $1000. 

4.  On  $2000  at  4i  mills  on  the  dollar. 

5.  On  $4000  at  52  cents  on  $100. 

6.  On  $10,000  at  1.4963%. 

7.  A  man's  tax  bill  at  1|%  is  $30.  What  is  the 
assessed  value  of  his  property  ? 

8.  What  is  the  actual  value  of  property  assessed  at 
$2000,  which  is  40%  of  its  actual  value  ? 

9.  At  how  many  mills  on  the  dollar  must  property  be 
taxed  to  raise  $  3000,  the  assessed  value  being  $  150,000  ? 


6o  Arithmetic 

10.  If  my  tax  is  $  6.30  at  the  rate  of  7  mills  on  the  dol- 
lar, what  is  the  assessed  value  of  my  property  ? 

11.  Find  the  tax  on  property  assessed  at  $2000,  the  rate 
being  $  11.50  per  $1000. 

12.  The   tax   on   property   assessed   at   $3000   is    $13. 
What  is  the  rate  ? 

13.  Find  the  collector's  fee  at  1%  on  the  tax  collected 
on  property  assessed  at  $  10,000,  the  tax  rate  being  1  % . 

14.  What  is  the  assessed  value  of  the  property  of  a  person 
whose  tax,  at  the  rate  of  5^  mills  on  the  dollar,  is  $  55  ? 

15.  What  rate  per  cent  is  equal  to  the  rate  of  4  mills  on 
a  dollar  ? 

108.    Written  Exercises. 

1.  A  tax  of  7.V  mills  on  a  dollar  was  levied  to  build  a 
school  costing  $3843.  What  was  the  assessed  value  of  the 
property  ? 

2.  What  is  the  tax  on  $  14,600  at  6f  mills  on  a  dollar  ? 

3.  If  the  rate  of  taxation  is  If  %,  find  the  tax  paid  by 
a  company  whose  property  is  assessed  for  $  105,000. 

4.  In  a  certain  town  the  tax  rate  is  8J  mills  on  the  as- 
sessed value  of  the  property.  If  property  is  assessed  75% 
of  its  actual  value,  what  will  be  the  tax  paid  on  property 
worth  $25,000? 

5.  Find  the  tax  on  a  house  worth  $15,600,  if  it  is  as- 
sessed at  f  of  its  actual  value  and  the  tax  rate  is  $12.80  on 
$1000. 

6.  A  town  having  an  assessed  valuation  of  $  5,600,000 
has  to  raise  by  taxation  $  84,000.     What  is  the  tax  rate  ? 

7.  A  school  district  wishes  to  raise  by  ta,xation  $47,500 
'for  a  new  schoolhouse.  If  the  assessed  value  of  the  prop- 
erty in  the  district  is  $2,500,000,  what  is  the  tax  rate? 


Taxes  and  Duties  6i 

8.  My   tax   last   year   on  an  assessment  of  $9200  was 
$80.50.     Find  the  rate  on  $  1000. 

9.  If  a  piece  of  property  is  taxed  $13.75  at  the  rate  of 
1%,  what  is  its  assessed  value  ? 

10.  If  the  tax  rate  is  6.7  mills,  what  is  the  assessed  value 
of  a  paper  mill  on  which  a  tax  of  $808.02  is  paid  ? 

DUTIES. 

109.  The  income  of  the  national  government  is  derived 
chiefly  from  two  sources : 

1.  Internal  revenue,  obtained  from  a  tax  on  liquors,  manu- 
factured tobacco,  and  certain  other  articles. 

2.  Duties  on  imports. 

110.  Duties  are  of  two   kinds :  specific  and  ad  valorem. 
The  former  is  a  certain  rate  per  pound,  yard,  gallon,  etc. 
The  latter  is  a  per  cent  of  the  value  of  the  goods  at  the. 
place  of  production.     Some  articles  pay  both  a  specific  and 
an  ad  valorem  duty. 

111.  Duties  are  paid  to  collectors  appointed  by  the  gov- 
ernment to  receive  them.  Their  offices  are  called  custom- 
houses and  are  located  at  ports  of  entry. 

112.  The  importer  delivers  to  the  collector  a  statement  of 
the  value  of  the  goods,  accompanied  by  the  invoice,  or  bill, 
rendered  him  by  the  foreign  seller.  The  value  is  given  in 
the  currency  of  the  country  from  which  the  goods  are  received 
and  is  verified  in  this  country  by  appraisers,  who  also  indi- 
cate the  rate  of  duty  to  be  paid. 

Goods  subject  to  specific  duties  are  weighed,  measured, 
or  gauged  by  government  officers. 

113.  The  base  in  ad  vcdorem  duties  is  the  foreign  value; 
the  duty  is  the  percentage. 


62  Arithmetic 

114.  Oral  Exercises. 

1.  Find  the  duty  on  an  automobile  valued  at  f  3000,  the 
rate  being  45%. 

2.  At  30^  per  bushel,  what  will  be  the  duty  on  250 
bushels  of  barley  imported  from  Canada? 

3.  Last  year  Christmas  trees  were  imported  to  the  value 
of  $2600.     What  duty  was  paid  at  20%  ? 

4.  Find  the  duty  on  400  bunches  of  shingles,  containing 
250  shingles  each,  at  30/  per  1000. 

5.  An  importer  of  toys  paid  $  105  duty.  What  was  the 
value  of  the  toys,  the  rate  being  35%  ? 

6.  Find  the  duty  at  1.44  cents  per  pound  on  2000 
pounds  of  sugar. 

7.  On  sugar  imported  from  the  Philippines  the  duty  is 
75  %  of  the  regular  rate.  Find  the  duty  on  1000  pounds, 
the  regular  rate  being  1.44  cents  per  pound. 

8.  Find  the  duty  on  500  pounds  of  Cuba  sugar  at  1.44 
cents  per  pound,  less  a  reduction  of  20%. 

9.  What  is  the  duty  on  100  square  yards  of  Brussels 
carpet  valued  at  $  120,  the  rate  being  44  cents  a  square  yard 
and  40%? 

10.    Find  the  duty  on  100  square  yards  of  dress  goods 
worth  14  cents  a  square  yard,  at  7  cents  a  square  yard  and 

50%. 

115.  Written  Problems. 

1.  A  dealer  imported  40  shotguns  at  $4  each  and  5  at 
f  60  each.  What  was  the  duty  at  $1.50  each  and  15%  on 
the  former,  and  $6  each  and  35%  on  the  latter? 

2.  Find  the  duty  on  400  yards  of  Brussels  carpet  f  yard 
wide,  valued  at  $1.25  per  running  yard,  the  rate  being  44 
cents  per  square  yard  and  40%  ad  valorem, 


Stocks  and   Bonds  6^ 

3.  Forty-six  million  pounds  of  beet  sugar  were  imported 
last  year.  What  sum  was  received  in  duties  at  1.615  cents 
per  pound  ? 

4.  What  per  cent  of  the  cost  is  paid  on  pearl-handled 
table  knives  valued  at  $3  per  dozen,  the  rate  being  16  cents 
each  and  15%  ? 

5.  Find  the  duty  at  1.^^  each  and  15%,  on  100  dozen 
table  knives  valued  at  50  ^   per  dozen. 

6.  An  importer  pays  a  duty  of  20  j^  each  and  40%,  on 
penknives  costing  him  $4  per  dozen  in  England.  At  what 
price  per  dozen  must  he  sell  them  to  make  a  profit  of  25% 
above  the  cost  in  this  country  after  the  duty  is  paid  ? 

7.  Snowshoes  to  the  value  of  $7099.65  were  imported 
last  year.     What  was  the  duty  at  20%  ? 

8.  From  Cuba  4128  pounds  of  starch  were  imported,  the 
duty  being  li  cents  per  pound,  less  20%.  How  much  duty 
was  received  ? 

9.  What  is  the  duty  at  1  cent  per  cubic  foot  on  100  hewn 
timbers,  each  16  feet  long,  2  feet  wide,  and  1  foot  thick? 

10.  AVhat  would  be  the  total  cost,  including  duty,  of  100 
pieces  of  oil  cloth,  each  20  yards  long  and  3  yards  wide,  at 
25^  per  square  yard,  the  duty  being  8j^  per  square  yard  and 
15%? 

STOCKS    AND    BONDS. 

STOCKS. 

116.  When  a  very  large  amount  of  money  is  required  to 
undertake  and  carry  on  a  business  enterprise,  a  stock  company, 
or  corporation,  is  generally  organized. 

117.  The  capital  stock  of  a  corporation  consists  of  the 
amount  invested,  which  is  divided  into  shares  of  stock, 
generally  of  $100  each.  To  each  stockholder  is  issued  a 
certificate  signed  by  the  proper  officers,  which  specifies  the 
number  of  shares  he  owns  and  the  original  value  of  the  share. 


64 


Arithmetic 


The  following  model  shows  the  usual  wording  of  a  stock 
certificate  : 


f ^  (h  i%  A  '<t^^  ^K  fh  i0 

IfJCORPORATED    UNDER   THE   LAWS   OF   THE   STATE   Of    ILLINOIS 


H    -^o-    "i^io 


150  Shares,  f^ 


^      (Eentral  JSanqfacturing  Sompany 


•^     ^bis  Certifies  that   ^^.'TfmJWiAvm^-  ..,^ 

"^^  is  the  owner  of  Qri£^  Hxi/ruinMi  J'lltnJ/    shares  of  fjf 


t:-^ 


^v    One  Jfainlred  Dollars  each  of  the  faU-]>aJd  stoett   .^^ 


•%^  o/  Mf  Central  Alanufaetarinfi  Contpanif. 

C: 


•i^-t-  I  raiisftTablc  oiilv  on   the  books  ot   llic  coiniiiinv    in   person   or  by     JS_«>. 

£^    (itlorney  itpou  surrender  of  this  eenilicale. 


-If- 


^onddTeftF 

Presideii 

yfi§%  t  If  I  If  1 1?  1 1?  I  %%  \\%\l  M  lit  lit  \  l'4  [  I?  \  |«  I  I?  ?  s?  I  S'?  I  |k  I  |i§  i"  %%  I  ITS.* 


President,    t^ 


118.  When  the  enterprise  shows  a  profit,  the  directors 
determine  what  portion  of  the  profits  should  be  retained  to 
extend  and  improve  the  business,  and  distribute  the  remain- 
der among  the  stockholders  as  a  dicidend.  The  dividend 
is  generally  fixed  as  a  certain  per  cent  of  the  capital  stock. 
Dividends  may  be  paid  annually,  semi-annually,  or  oftener. 

119.  The  original  value  of  stock  named  in  the  certificate 
is  called  the  face  or  par  value.     When  the  corporation  is 


Stocks  and  Bonds  6^ 

paying  satisfactory  dividends,  the  stock  generally  sells 
above  par;  if  the  profits  are  small  or  nothing, the  stock  sells 
below  par. 

120.  Stocks  are  generally  bought  and  sold  through  brok- 
ers, who  receive  a  brokerage  (commission)  of  $12.50  for 
each  purchase  or  sale  of  100  shares  of  stocks  of  the  par 
value  of  f  100  each,  regardless  of  the  price  actually  paid. 
This  amounts  to  ^%  of  thenar  value.  Thus,  a  broker  sell- 
ing 100  shares  of  Baltimore  &  Ohio  for  $12,200  would 
receive  the  same  brokerage,  $12.50,  as  would  be  paid  him 
for  the  sale  of  100  shares  of  Erie  for  $4425. 

121.  Oral  Problems. 

1.  A  company  is  incorporated  with  a  capital  of  $150,000. 
How  many  shares  of  $100  each? 

2.  What  per  cent  of  the  capital  stock  of  $100,000  is 
held  by  the  owner  of  25  shares  ? 

Note.  If  no  other  value  is  given,  tlie  par  value  is  assumed  to  be 
$100. 

3.  If  a  dividend  of  3%  is  declared,  how  much  is  received 
by  the  owner  of  40  shares  of  stock? 

4.  What  is  the  annual  income  of  the  owner  of  25  shares 
of  stock  on  which  is  paid  a  semi-annual  dividend  of  4%  ? 

5.  What  is  the  value  of  20  shares  of  Baltimore  &  Ohio 
selling  at  122-1- ? 

6.  What  must  be  paid  for  10  shares  of  N.Y.  Central  at 
119 J  and  the  brokerage  of  i%  ? 

7.  If  I  pay  $125  for  a  share  of  stock,  and  receive  $5 
dividends  annually,  what  rate  per  cent  do  I  receive  annually 
on  my  investment? 

8. .  I  pay  $  150  for  a  $  100  share  of  stock  and  receive  in 
dividends  4%  annually  on  my  investment.  What  per  cent 
of  the  par  value  is  the  dividend  ? 


66  Arithmetic 

9.  What  per  cent  dividend  must  be  declared  in  order 
that  I  may  receive  4%  on  the  sum  I  have  paid  for  stock 
that  cost  me  250  ? 

10.  I  receive  a  semi-annual  dividend  of  4%  on  Adams 
Express  Co.  stock  for  which  I  paid  300.  What  per  cent  do 
I  receive  annually  on  my  investment? 

Written  Exercises. 

122.  Find  the  cost  to  the  purchaser,  adding  i%  broker- 
age in  each  case: 

1.  20  shares  of  Delaware,  Lackawanna,  &  Western,  par 
value  $  50  each,  at  465. 

$50  X  20  X  4.651  -  $ 4651.25.  Ans.  The  par  value  of  20  shares  is 
$50  X  20,  or  $  1000.  The  price,  including  the  brokerage,  is  465^%  of 
the  par  value,     f  1000  x  4.651  gives  the  total  cost. 

To  find  the  cost  of  stock,  tnultiply  the  par  value  of  the  stock 
by  the  price  {including  the  brokerage)  expressed  as  a  per  cent. 

Note.  The  par  value  is  taken  at  f  100  per  share,  unless  otherwise 
specified. 

2.  100  shares  of  Pennsylvania  R.R.,  $50  each,  at  119f. 

3.  60  shares  of  Toledo,  St.  Louis,  &  Western,  at  28. 

4.  100  shares  of  National  Lead  Co.,  at  59|. 

5.  10  shares  of  Canadian  Pacific,  at  166J. 

6.  250  shares  of  Iowa  Central,  at  40. 

Find  the  amount  received  by  the  seller,  after  the  deduc- 
tion of  the  brokerage  of  |  %  : 

7.  72  shares  of  Amalgamated  Copper  at  841. 

$100  X  72  X  .84|  =  $6084.  Ans.  The  par  value  of  72  shares  is 
$  7200.  Price,  84f %  -  \  =  84i%,  deducting  brokerage.  $  7200  x  .84^ 
gives  the  amount  paid  over  to  the  seller. 

8.  175  shares  of  Chicago  &  Alton,  at  16. 

9.  450  shares  of  Corn  Products  Eefining  Co.,  at  75J. 
10.    20  shares  of  Illinois  Central,  at  137. 


Stocks  and  Bonds  67 

123.  Find  the  per  cent  realized  upon  an  investment  in 
American  Car  &  Foundry  Co.  at  39 J,  dividends  ^  %  quarterly. 

Assuming  that  the  stock  is  bought  through  a  broker,  its  cost  will 
be  39|%  +  i  %,  or  40  %  of  the  par  value.  The  annual  dividend  is  2  %  of 
the  par  value ;  that  is,  the  owner  of  a  one-hundred-dollar  share  re- 
ceives !?2  annual  dividend  on  his  investment  of  $40.  The  problem 
resolves  itself  into  finding  what  per  cent  $  2  is  of  $  40. 

Rate  =  2%  ^  .40  =  5%.  Ans. 

.  To  find  the  rate  yielded  by  a  stock  investment,  divide  the 
annual  rate  of  dividend  by  the  cost  {including  brokerage^  con- 
sidered as  a  per  cent. 

124.  Find  the  rate  yielded  by  an  investment  in  the  fol- 
lowing: 

Note.     Add  \  %  brokerage  to  the  price  of  each. 

1.  Western  Union  Telegraph  at  79J,  quarterly  dividend 

2.  Great  Northern,  at  124 J,  quarterly  dividend  1}  % . 

3.  Chesapeake  &  Ohio  at  37|-,  annual  dividend  1  %. 

4.  People's  Gas  at  89|-,  quarterly  dividend  1|  %. 

5.  Rock  Island,  at  49 J,  annual  dividend  1  %. 

125.  Written  Problems. 

1.  A  broker  sold  for  Mr.  Freeman  285  shares  of  Dela- 
ware &  Hudson  at  187^.  What  did  Mr.  Freeman  receive  ? 
Find  the  broker's  commission. 

2.  Canada  Southern  stock  is  bought  for  me  at  601 
The  broker's  bill  is  f  7230,  which  includes  his  commission 
of  i%.     How  many  $100  shares  does  he  buy  for  me  ? 

3.  Mr.  Curran  received  from  his  broker  $10,174.50  as 
the  proceeds  of  Union  Pacific  stock  sold  at  134.  What  was 
the  par  value  of  the  shares  ? 


68  Arithmetic 

4.  John  Griffin  paid  f  9330  for  80  shares  of  Consoli- 
dated Gas  bought  through  his  broker.  What  rate  did  the 
broker  pay  ? 

5.  A  man  bought  25  shares  of  Union  Pacific  at  1481 
and  sokl  them  at  146|^.  What  was  his  loss,  including  i% 
each  for  buying  and  for  selling  ? 


BONDS. 

* 

126.  When  the  United  States  government,  a  statev,  or  a 
city  wishes  to  borrow  money,  it  issues  bonds,  which  are 
agreements  to  pay  at  a  stated  time  a  certain  sum  of 
money,  and  a  certain  percentage  thereon  annually,  semi- 
annually, or  quarterly,  until  the  redemption  of  the  bonds. 
The  money  paid  for  the  use  of  the  borrowed  money  is 
called  interest. 

127.  In  addition  to  sums  raised  by  the  sale  of  stock, 
many  corporations  obtain  money  by  the  issue  of  bonds. 
The  payment  of  the  bonds  is  frequently  secured  by  pro- 
viding that  the  holders  may  sell  the  property  of  the  cor- 
poration if  default  should  be  made  in  the  payment  of  the 
face  value  of  the  bonds  as  they  become  due,  or  in  the  pay- 
ment of  the  interest  instalments. 

128.  The  holders  of  stock  in  a  corporation  are  the 
owners ;  the  holders  of  bonds  are  persons  to  whom  the 
stockholders  are  indebted  for  money  loaned.  The  stock- 
holders receive  dividends  that  vary  according  to  the  profits 
of  the  corporation.  The  bondholders  receive  a  fixed  rate 
on  the  money  loaned,  and  the  principal  sum  when  the  bonds 
are  due.  When  this  is  paid,  the  bonds  cease  to  exist ;  the 
stock  continues  while  the  corporation  lasts.  Stock  is  issued 
in  shares  of  the  same  par  value  for  each  share ;  bonds  are 
issued  of  different  par  values;  say  of  $  100,  $  500,  $  1000,  etc. 


Stocks  and  Bonds  69 

Note.  Bonds  are  described  as,  for  instance,  Illinois  Central  4's, 
98 1 ;  which  means  that  the  bonds  of  the  Illinois  Central  Railroad  pay- 
ing 4  %  interest  sell  at  98|. 

129.  Oral  Exercises. 

Find  the  annual  income  ou  the  following  bonds : 

1.  Par  value  $3000;  rate  5%. 

2.  Par  value  $  5000 ;  rate  4  %. 

3.  Par  value  $  4000  ;  rate  6%. 

4.  Par  value  $  5000  ;  rate  5  %. 

5.  Par  value  $10,000;  rate  3.65  %. 

Find  the  cost  of  the  following  bonds,  adding  |-%  bro- 
kerage : 

6.  Par  value  $3000;  price  119f 

7.  Par  value  $4000  ;  price  87|. 

8.  Par  value  $1000;  price  99f. 

9.  Par  value  $2000;  price  123. 

10.  Par  value  $50  ;  price  841 

Find  the  proceeds  of  sales  of  the  following  bonds,  less 
brokerage : 

11.  Par  value  $3000  ;  price  120i 

12.  Par  value  $  2000  ;  price  87i 

13.  Par  value  $1000;  price  99f. 

14.  Par  value  $4000;  price  125i. 

15.  Par  value  $  1600  ;  price  87|. 

Written  Exercises. 

130.  1.    Find   the   cost   of    bonds   of   the   par  value   of 
$7000  bought  at  118|,  brokerage  ■}%. 

Cost  =  $  7000  X  (1.18f  +  .001). 
To  find  the  cost  of  bonds,  midtiply  the  par  value  by  the  price 
(including  the  brokerage)  expressed  as  a  p>er  cent. 


JO  Arithmetic 

Find  the  cost  of  the  following,  including  brokerage : 

2.  ^4000  Imperial  Japanese  4i's  at  92f. 

3.  $7000  U.  S.  of  Mexico  5's  at  98. 

4.  $  6000  Kepublic  of  Cuba  5's  at  1021 

5.  $  8000  Philippine  4's  at  111. 

6.  $5000  Wisconsin  Central  4's  at  85i. 

7.  Find  the  amount  received  for  $  6000  Northern  Pacific 
4's  sold  at  lOOf,  less  brokerage  at  i%. 

$6000  X  (l.OOf-.OOi). 

Find  the  amount  received  for  the  following,  less  bro- 
kerage : 

8.  $3500  Western  Union  4i's,  at  95. 

9.  $4500  Canada  Southern  5's,  at  lOlf 

10.  $2500  Kansas  City  Southern  3's,  at  69. 

11.  $1500  Lake  Shore  SJ^s,  at  93f. 

12.  $  10,000  Short  Line  4's,  at  88f . 

131.  1.  Find  the  annual  income  from  Albany  &  Sus- 
quehanna 3^'s,  par  value  $  7500. 

Par  value  x  rate  of  interest  =  annual  income 
$7500x.03| 

Find  the  annual  income  of : 

2.  Lake  Shore  3i's,  par  value  $  3300. 

3.  Lehigh  and  Wilkesbarre  4i's,  par  value  $4800. 

4.  Baltimore  and  Ohio  3|^'s,  par  value  $1750. 

5.  Hocking  Valley  4^'s,  par  value  $12,500. 

6.  District  of  Columbia  3.65's,  par  value  $8000. 

132.  In  ascertaining  the  rate  of  iiicome  produced  by  a  bond  bought 
eitlier  above  or  below  par,  consideration  is  taken  of  the  fact  that  par 
value  is  to  be  paid  when  the  bond  falls  due.  For  this  reason  the  rate 
of  income  actually  yielded  may  differ  materially  from  the  rate  speci- 
fied in  the  bond. 


;ate. 

Due. 

Price. 

Yielding. 

H 

Sept.  1927 

104.25 

4.20% 

H 

Oct.   1913 

102.00 

4.10% 

5 

Jan.  1917 

95.00 

5.40% 

Commercial  Discount  71 

The  calculation  of  this  actual  rate  of  income  involves  the  application 
of  the  principles  of  compound  interest,  in  a  manner  so  complicated 
that  it  cannot  be  treated  in  a  common  school  arithmetic. 

Bankers  and  brokers  use  for  this  purpose  carefully  prepared  tables, 
and  the  question  of  obtaining  the  actual  rate  of  income  yielded  by  a 
given  bond  should  generally  be  referred  to  these  tables. 

Quotations  as  above  were  recently  made  on  bonds  as  follows : 

Bonds. 
City  of  Omaha,  Nebraska, 
State  of  Tennessee, 
South  Dakota  Central  Railway, 
Northern  Illinois  Light  &  Trac- 
tion Co.,  6        July  1923        95.00      5.50% 


COMMERCIAL  DISCOUNT. 

133.  It  is  the  practice  of  many  manufacturers  of  certain 
classes  of  goods  to  issue  catalogues  of  their  products,  in 
which  they  quote  prices  higher  than  those  at  which  the 
goods  are  actually  sold.  The  price  printed  in  the  catalogue 
is  the  list  price.  Their  customers  receive  a  discount  sheet 
notifying  them  of  the  rate  of  discount,  a  new  sheet  being 
sent  at  each  fluctuation  of  the  actual  selling  price. 

List  prices  are  sometimes  known  as  catalogue  prices. 
Commercial  discounts  are  also  called  trade  discounts. 

134.  Two  or  more  discounts  are  frequently  given :  as,  70 
and  30%  ;  50  and  17^%  ;  60, 10,  and  10%,  etc.  One  of  these 
discounts  is  deducted  from  the  list  price  as  a  base;  the 
remainder  is  the  base  from  which  the  next  discount  is  de- 
ducted. Thus,  $200  less  70  and  30%  would  indicate  that 
f  200  was  reduced  by  70%,  or  $  140,  leaving  a  remainder  of 
$60,  and  that  this  remainder  was  to  be  reduced  30%,  or 
$  18,  leaving  a  net  price  of  $  42. 

Note.  The  mark  %  is  generally  written  only  after  the  last  of  a 
series  of  successive  discounts. 


72 


Arithmetic 


135.  It  is  usual  in  some  branches  of  the  wholesale  trade 
to  allow  a  discount  for  payment  made  before  the  expiration 
of  the  credit  time.  Thus,  a  bill  of  goods  amounting  to  $400 
payable  in  30  days  might  be  reduced  2%  by  payment  at 
the  time  of  purchase,  or  1%  by  payment  at  any  time  within 
15  days.  This  cash  discount  is  deducted  from  the  cost 
remaining  after  the  deduction  of  the  other  discounts. 

136.  The  following  bill  shows  the  method  of  stating 
trade  and  cash  discounts : 


Slatington,  Penn.,  Jan.  16,  1908. 
Mr.  Kichakd  Hadden 

Bought  of  The  Hugo  Katzler  Company 
Terms :  Cash  less  2%. 


100  gross  slates,  #14     $  8.75 

less  20% 

less  10% 

less    5%, 

Gash,  less    2% 

Keceived  payment, 
Jan.  16,  1908, 
Thei  Hugo  Katzler  Company, 
per  C.  B.  J.  Smythe. 

137.    Oral  Exercises. 
Find  the  net  prices : 

1.  List  price  $  80;  discounts  50  and  50% 

2.  List  price  $  100;  discounts  50  and  25% 

3.  List  price  $160;  discounts  25  and  10% 

4.  List  price  $  200 ;  discounts  10  and  10% 


875 

175 

700 

70 

630 

31 

50 

598 

50 

11 

97 

$586 

53 


Commercial   Discount  73 

5.  List  price  $  300 ;  discounts  33i  and  20% 

6.  List  price  $  800 ;  discounts  50  and  10% 

7.  List  price  $500;  discounts  20  and  10% 

8.  List  price  $400;  discounts  25  and  20% 

9.  List  price  $60;  discounts  16|  and  10% 
10.  List  price  $50;  discounts  80  and  80% 

138.    Written  Exercises. 

1.  Find  the  net  price  of  goods  listed,  or  catalogued,  at 
$484,  on  which  are  allowed  trade  discounts  of  25  and  10%, 
and  a  cash  discount  of  5%. 

$968  Businessmen  do  not  write  unnecessary  figures. 

less  I      242  25%  of  $968  is  obtained  by  dividin^r  it  by  4.     To 

$726  /u       ■«-                                 J               o          J 

,       j_      ^2  fio  divide  726  by  10,  the  numbers  composing  it  are 

^"^^.-o  .<r>  rewritten  one  place  to  the  right.     To  divide  by  20, 

less  J-      32.67  divide  by  2,  placing  each  quotient  figure  one  place 

NeV  $620.73  to  the  right. 


Find  the  net  prices: 

List  price.  Discounts.  List  price.         Discounts. 

2.  $1484  50  and  10%  7.  $1800   50, 10,  and  10% 

3.  $371.90  40  and  30%  8.  $2400   30, 10,  and   5% 

4.  $225.80  60  and  20%  9.  $4000    60,  25,  and  10% 

5.  $3075  70  and  30%  10.  $1700    10,10,  and  5% 

6.  $4500  25  and  20%  11.  $800     40,30,  and  10% 

139.    Written  Problems. 

1.  What   is   due   on   a   bill  of  hardware  amounting   at 
list  prices  to  $480,  the  discounts  being  40,  12i,  and  10%? 

2.  Find  the  difference  on  a  bill  of  $700  between  a  single 
discount  of  50%,  and  successive  discounts  of  25, 15,  and  10%. 

3.  Abraham  Stein  paid  $  330  in  settlement  of  a  bill  for 
corrugated  iron  pipe  amounting  at  list  prices  to  $800.    One 


74  Arithmetic 

of  the  two   successive  discounts  was  50%  ;  what  was  the 
other? 

4.  On  a  bill  of  goods  listed  at  $  100,  compare  discounts  of 
40,  25,  and  10%,  discounts  of  10,  25,  and  40%,  and  discounts 
of  25,  40,  and  10%. 

5.  A  buyer  pays  80%  of  90%  of  the  catalogue  price  of 
$  875  for  a  piano.  What  does  he  pay  for  it  ?  What  two 
successive  discounts  does  he  receive?  What  per  cent  of 
the  catalogue  price  does  he  pay  ?  What  equivalent  single 
discount  does  he  receive? 

6.  A  man  had  $630  after  spending  20%  of  his  money 
and  then  10%  of  the  remainder.  What  sum  had  he  at 
lirst? 

7.  Make  out  a  bill  for  250  dozen  penknives  sold  by 
George  Freiberg  to  Nicholas  J.  Barnett  at  $12  per  dozen 
less  25  and  10%,  and  5%  for  cash. 

8.  lam  offered  40  and  10%  by  one  dealer  and  30  and 
20%  by  another.  Which  should  I  accept,  and  what  would  I 
save  on  a  purchase  of  goods  listed  at  $275? 

9.  A  dealer  marks  an  article  50%  above  $80,  its  cost, 
and  sells  it  at  a  discount  of  30%  from  the  marked  price. 
What  per  cent  does  he  gain? 

10.  What  per  cent  would  be  lost  by  selling  an  article  for 
$90,  which  was  40%  below  the  marked  price,  the  latter 
being  50%    above  cost? 

140.   Preliminary  Exercises. 

1.  A  man  has  $  100.  How  much  will  he  have  left  after 
spending  ^  of  it  and  J^  of  the  remainder  ? 

2.  What  fraction  of  his  money  has  a  man  after  spending 
T^ofit? 

3.  If  a  man  has  -^^  of  a  certain  sum  and  spends  ^^  of 
this  ^^f  what  fraction  of  this  -^^  remains  ? 


Commercial   Discount  75 

4.  What  fraction  of  a  man's  money  equals  -f^  of  ^^  of  his 
money  ? 

5.  What   fraction  of   a   man's  money  is   left  after  he 
spends  y\  of  it  and  ^^  of  the  remainder? 

6.  Find  the  value  of  .(1  -  j\)  X  (1  -  yV)- 

7.  What  per  cent  is  j%  of  70  %  ? 

8.  90  %  of  70  %  equals  what  per  cent  ? 

9.  30  and  10  %  discount  equals  what  per  cent  net  ? 

The  deduction  of  the  first  discount  of  30%  leaves  a  remainder  of  70 
per  cent,  90%  of  which  gives  a  net  price  of  63  per  cent  of  the  list  price. 
Ans.  63%- 

141.    Oral  Exercises. 

Find  the  per  cent  remaining  after  the  deduction  of  the 
following  discounts : 

1.  50  and  50  %  6.   20  and  20  % 

2.  40  and  40  %  7.   10  and  10  % 

3.  60  and  10  %  8.    50,  20,  and  10  % 

4.  70  and  30  %  9.   30,  20,  and  121  % 

5.  60  and  20  %  10.   33i,  10,  and  10  % 

11.  W^hat  single  discount  equals  successive  discounts  of 

30  and  10  %  ? 

The  first  discount  is  30  % 

The  second  discount  is  J^  of  70  %  7% 

Total  discount  37  % 

Find  the  single  discount  equivalent  to  each  of  the  fol- 
lowing : 

12.  50  and  50  %  17.  20  and  20  % 

13.  40  and  40  %  18.  10  and  10  % 

14.  60  and  10  %  19.  50,  20,  and  10  % 

15.  70  and  30  %  20.  30,  20,  and  12^  % 

16.  60  and  20  %  21.  33i,  10,  and  10  % 


CHAPTER   III. 

APPLICATIONS   OF  PERCENTAGE  INVOLVING  THE 
ELEMENT   OF   TIME. 

INTEREST. 

142.  A  person  in  need  of  money  for  any  purpose  may 
obtain  it  by  furnishing  proper  security  for  its  repayment  at 
a  specified  time,  and  by  agreeing,  in  return  for  the  use  of 
the  money,  to  pay  a  certain  per  cent  of  the  sum  loaned. 

143.  The  sum  paid  for  the  use  of  money  is  called  interest. 
The  sum  borrowed  is  called  the  principal.  The  rate  is  the 
per  cent  agreed  upon  for  the  use  of  the  money  for  a  year. 
The  amount  is  the  principal  plus  the  interest. 

144.  Oral  Exercises. 

Find  the  interest  on  $  800  for  one  year  at : 

1.  3%                        5.    21%  9.  41% 

2.  4%                         6.    31%  10.  41% 

3.  5%                        7.   31%  11.  4f% 

4.  6%                        8.    3f%  12.  51% 

Find  the  yearly  interest  at  4  %  on : 

13.  $200                   17.    $250  21.  $275 

14.  $400                   18.    $350  22.  $375 

15.  $600                   19.    $450  23.  $475 

16.  $800  20.    $550  24.    $575 

76 


Interest  77 

145.  Find  the  yearly  interest  on  : 

1.  $300  at  4%  6.  $600  at  41% 

2.  $400  at  5%  6.  $200  at  51% 

3.  $500  at  6%  7.  $400at3f% 

4.  $100  at  31%  8.  $800  at  81% 

Find  the  interest  at  6  %  on  : 

9.    $100  for  3  years  13.  $50  for  4  years 

10.  $250  for  2  years  14.  $80  fori  year 

11.  $400  for  11  years  15.  $70  for  6  months 

12.  $300  for  21  years  16.  $  120  for  4  months 

146.  In  calculating  interest  the  general  rnle  is  to  con- 
sider the  year  as  consisting  of  12  months  of  30  days  each. 

147.  General  Method. 

1.    Find  the  interest  on  $350  for  3  years  6  months,  at 

6%. 

Principal     §  350 

Rate         .06 

Interest  for  1  year  .$21.00 

Time  in  years    3^ 

$10.50 

63.00 

Interest  for  3  yr.  6  mo.        .$  73.50 

Find  the  interest  for  a  year  by  multiiJlyhig  the  principal  by 
the  rate  considered  as  hundredths.  Multijyly  the  interest  for  one 
year  by  the  number  of  years. 

Find  the  interest  on  :       * 

2.  $  1628  at  4%  for  3  years 

3.  $2156at  5%  for  9  years 

4.  $2326  at  3%  for  7  years 


78  Arithmetic 

m 

5.  f  1668  at  3%  for  8  years 

6.  ^620  at  5%  for  4  years 

7.  $262.50  at  3%  for  6  years 

8.  $346.75  at  4%  for  4  years 

9.  $679.40  at  5%  for  6  years 

10.  $548  at  5i%  for  2  years 

11.  $2166  at  41%  for  3  years 

12.  $325.20  at  5%  for  6  years 

13.  $628.40  at  4%  for  6  years 

14.  $  769.10  at  3%  for  7  years 

15.  $878.60  at  4%  for  4  years 

•    16.  $340  at  31%  for  2  years  6  months 

17.  $584.50  at  41%  for  3  years  3  months 

18.  $736.40  at  3|%  for  4  years  9  months 

19.  $825.60  at  4i%  for  3  years  8  months 

20.  $1243.20  at  5^%  for  4  years  4  months 

148.    Cancellation  Method. 

1.  Find  the  interest  on  $1875  for  2  years  7  months 
21  days  at  4%. 

2  years  =  720  days  $^               317 

7  months  =  210  days  ^  ^±^m^njS5^ 

21  days  =    21  days  X^<^     3^0           8                     ^ 

2  yr.  7  mo.  21  da.  =  951  days  ^^       ^^ 

=    951    yj.  " 

^'^•^  '  ^ws.  $198.13- 

This  method  is  frequently  found  convenient  when  the  time  contains 
days,  the  whole  number  of  days  being  found  and  placed  above  360,  as 
a  denominator,  to  reduce  the  time  to  years.  The  rate  is  generally  ex- 
pressed as  a  fraction  :  4i %  as  gg^  ;  3|  %  as  ^^^q,  etc.  The  operations 
are  indicated  without  reducing  any  fractions  to  lower  terms,  the 
shortening  of  the  work  by  cancellation  accomplishing  the  same  result. 


Interest  yp 

Find  the  interest  and  the  amount : 

2.  f  1300  at  3%  for  '72  days 

3.  $1700  at  5%  for  100  days 

4.  $  5200  at  2%  for  2  months  6  days 

5.  $840  at  4%  for  16  days 

6.  $136  at  6%  for  8  months  8  days 

7.  $240  at  U%  for  13  months  20  days 

8.  $480  at  2i%  for  1  yr.  1  mo.  20  da. 

9.  $360  at  3|%  for  2  yr.  4  mo.  15  da. 
10.  $1800  at  6%  for  3  yr.  7  mo.  21  da. 

149.  When  the  time  between  two  given  dates  is  less  than 
a  year,  it  is  usual  in  calculating  interest  to  ascertain  the 
exact  number  of  days  between  the  two  dates. 

1.  Find  the  interest  on   $685  at  5%   from  July  1  to 
September  23. 

The  time  remaining  in  the  first  month  is  found  by  subtracting  the 
given  date  from  the  number  of  days  in  the  month.  To  this  are  added 
the  whole  number  of  days  in  the  intervening  month  and  the  number 
of  days  expressed  by  the  date  of  the  last  month. 

July  1  to  July  31,  30  days  $6.85       „        J       ^,.  q. 

August,  31     "  $^^^x -^x -^=^ii:^  =  $7.99  + 

In  September,  23     '-  ^^^      ^^^  ^ 

84  days  q 

Ans.  |!7.99 

Note.  The  100  in  a  denominator  is  generally  canceled  by  chan- 
ging the  decimal  point  in  a  numerator. 

Find  the  interest  and  the  amount : 

2.  $420  at  3^%  from  March  6  to  May  2 

3.  $360  at  U%  from  August  16  to  October  23 

4.  $540  at  4%  from  May  16  to  November  13 

5.  $680  at  6%  from  March  23  to  July  7 


8o  Arithmetic 

G.  $720  at  5%  from  June  28  to  August  16 

7.  $1080  at  21%  from  Dec.  12,  1907,  to  Feb.  29,  1908 

8.  $1440  at  4%  from  Nov.  11,  1907,  to  March  24,  1908 

9.  $960  at  6%  from  Oct.  10,  1907,  to  Feb.  25,  1908 
10.  1640  at  5%  from  July  17,  1907,  to  April  6,  1908 

150.  When  the  dates  are  more  than  a  year  apart,  the 
time  is  found  by  ascertaining  first  the  number  of  years,  then 
the  number  of  months,  then  the  number  of  days. 

1.  Find  the  amount  of  $765  at  4i%  from  Jan.  17,  1908, 
to  May  3,  1910. 

The  time  from  Jan.  17,  1908,  to  Jan.  17,  1910    is  2  yr.,    or   720  da. 
The  time  from  Jan.  17,  1910,  to  April  17,  1910  is  3  mo.,  or     90  da. 

The  time  from  April  17,  1910,  to  May  3,  1910     i^ 16  da. 

The  time  from  Jan.  17,  1908,  to  May  3,  1910      is  826  da. 

$.765 

$7.^^  413 

Interest  =  $7^^  x  ^  x  ^  ^  $315,945  ^  ^^3  qq  _ 
%^^     3^0  4 

4P 

Amount  =  $  765  +  $  78.99  =  $  843.99.  A ns. 

Find  the  amount : 

2.  $300  at  6%  from  Dec.  27,  1907,  to  Oct.  20,  1909 

3.  $400  at  5%  from  July  16,  1907,  to  May  10,  1910 

4.  $500  at  4%  from  Sept.  14,  1907,  to  July  8,  1911 

5.  $600  at  4i%  from  Apr.  20,  1907,  to  Dec.  5,  1912 

151.  Method  by  Aliquot  Parts. 

1.    Find  the   interest    on   $247.56  at  4%   for  3   years 
8  months  27  days.     Find  the  amount. 

All  the  work  necessary  to  obtain  tlie  result  is  given  here.  Under 
the  principal  is  written  the  interest  for  a  year,  by  multiplying  by  4 
and  writing  the  first  figure  of  the  product  two  places  to  the  right. 
The  interest  for  the  remaining  2  years  is  next  found.     The  interest 


Interest  8i 

for  8  months  is  obtained  by  dividing  by  3  the  interest  for  2  years.  The 
interest  for  24  days  is  j^q  of  the  interest  for  8  months,  or  240  days. 
The  interest  for  3  days  is  I  of  the  interest  for  24  days.  Adding  to- 
gether the  interest  items,  we  get  the  interest.  By  including  the  prin- 
cipal, we  obtain  the  amount. 

Principal  $247.56 

Interest  for  1  yr. 
Interest  for  2  yr. 
Interest  for  8  mo. 
Interest  for  24  da. 
Interest  for  3  da. 

Interest  for  3  yr.  8  mo.  27  da.  •$  37.05  +   Ans. 
Amount  •?  284.61  +  Ans. 

2.  Find  the  amount  of  $247.56  at  4i%  for  3  years  8 
months  27  days. 

The  interest  for  2  years  is  first  found,  employing  9  as  a  multiplier 
instead  of  4i,  which  would  be  used  to  obtain  the  interest  for  1  year. 
Aliquot  parts,  other  than  those  used  in  the  previous  example,  are 
employed  to  show  that  variety  is  possible. 

Principal  $247.56 


$  9.9024 

19.8048 

6.6016 

^  of  int.  for  2  yr. 

.6602 

i\j  of  int.  for  8  mo, 

.0823 

i  of  int.  for  24  da. 

Interest  for  2  yr.  ■» 
Interest  for  1  yr.  J 

22.2804 

11.1402 

1  of  2  yr. 

Interest  for  6  mo.  ■> 
Interest  for  2  mo.  J 

5.5701 

i  of  1  yr. 

1.8567 

i  of  6  mo. 

Interest  for  18  da.  -» 
Interest  for  9  da.    j 

.5570 

Jq  of  6  mo. 

.2785 

1  of  18  da. 

Amount  $289.24  4-   A 

ns. 

3.    Find  the  interest  on  $  125  at  6% 

for  1  yr.  11  mo.  14 

da. 

Interest  for  1  year 

$7.50 

Interest  for  6  months 

^  of  $  7.50 

Interest  for  4  months 

1  of  $  7.50 

Interest  for  1  month 

1  of  int.  for  4  mo. 

Interest  for  12  days 

Jj5  of  int.  for  4  mo. 

Interest  for  2  days 

1  of  mt.  for  12  da. 

82  '  Arithmetic 

4.  Find  the  amount  of  $240  at  5%  for  3  yr.  5  mo.  18  da. 

Principal  $  240. 

Interest  for  2  yr.        24. 
Jnterest  for  1  yr. 
Interest  for  4  mo. 
Interest  for  1  mo. 
Interest  for  15  da. 
Interest  for  .3  da. 

5.  Find  the  amount  of  ^720  at  5^%   for  4  yr.  7  mo. 

21  da. 

Principal  $  720. 


Interest  for  2  yr. 

79.20 

11%  of  principal 

Interest  for  2  yr. 

79.20 

Interest  for  6  mo. 

' 

I  of  int.  for  2  yr. 

etc. 

etc. 

6.    Find  the  interest  on  $124.75  at  7%  for  2  yr.  6  mo. 

18  da. 

1  yr.,  1  yr.,  6  mo.,  18  da. 

152.  When  the  time  is  less  than  a  year,  and  the  rate  is 
6%,  advantage  is  taken  of  the  fact  that  the  interest  for  60 
days  is  1%  of  the  principal. 

1.   Find  the  amount  of  $248  at  6%  for  208  days. 


Principal 

$248. 

Interest  for  60  da. 

2.48 

1%  of  principal 

Interest  for  120  da. 

4.96 

twice  int.  for  60  da. 

Interest  for  20  da. 

.8267 

^  of  int.  for  60  da. 

Interest  for  6  da. 

jL  of  int.  for  60  da. 

Interest  for  2  da. 

i  of  int.  for  6  da. 

2.  Find  the  interest  on  $  1256  at  6%  for  66  days. 

60  da.  +  6  da. 

Find  the  interest  at  6%  : 

3.  $  1873  for  84  days.  6.  $  249  for  219  days. 

4.  $956.50  for  167  days.  7.  $157.40  for  175  days. 

5.  $48.76  for  315  days.  S.  $863.20  for  93  days. 


Interest  83 

Find  the  amount,  the  rate  being  6% : 

9.  $  1200  from  Feb.  1,  1909,  to  Aug.  15,  1909. 

10.  $1200  from  Mar.  8,  1908,  to  Sept.  14,  1908. 

11.  $1200  from  Jan.  9,  1907,  to  Dec.  1,  1907. 

12.  $1200  from  Apr.  15,  1908,  to  June  26,  1908. 

13.  $1200  from  May  25,  1909,  to  Nov.  19,  1909. 

14.  $1200  from  July  30,  1908,  to  Oct.  29,  1908. 

153.  Interest  is  the  product  of  three  factors  :  the  princi- 
pal, the  rate,  and  the  time.  By  the  ordinary  method  the 
interest  is  calculated  for  a  year,  and  this  result  is  multiplied 
by  the  number  of  years ;  that  is. 

Interest  for  given  time  =  interest  for  1  year  x  number  of 
years. 

154.  Some  examples  are  more  readily  worked  by  multi- 
plying the  principal  by  the  rate  for  the  given  time ;  that  is, 

Interest  for  given  time  =  principal  x  rate  for  given  time. 

155.  Preliminary  Exercises. 

Find  the  rate  :  * 

1.  For  60  days,  at  6%  per  year. 

2.  For  4  years,  at  5%.  9.  For  144  days,  at  5%. 

3.  For  2  years,  at  41%.  10.  For  288  days,  at  0%. 

4.  For  4  years,  at  of  %.  11.  For  36  days,  at  5%. 

5.  For  1  month,  at  6%.  12.  For  1  year,  at  ^%. 

6.  For  i  year,  at  5%.  13.  For  40  days,  at  41%. 

7.  For  72  days,  at  5%.  14.  For  80  days,  at  4^%. 

8.  For  216  days,  at  5%.  15.  For  160  days,  at  ^%. 


8+ 


Arithmetic 


156 

.    Sight  Exercises. 

# 

Find  the  interest: 

1. 

Rate  6%  ; 

time 

60  days; 

principal  $  146 

1  %  of  the  principal. 

2. 

Rate  5%  ; 

time 

2  years; 

principal  $430 

^^  of  the  principal. 

3. 

Rate  41%; 
9%  of  the  prin 

time 
3ipal. 

2  years; 

principal  $109 

4. 

Rate  3f%; 

time 

4  years; 

principal  $100 

15%  of  the  principal. 

5. 

Rate  6%; 

time 

1  month; 

principal  $184 

1  %  of  the  principal. 

6. 

Rate  5%; 

time 

i  year; 

principal  $173 

7. 

Rate  5%; 

time 

72  days; 

principal  $217 

8. 

Rate  5%; 

time 

216  days; 

principal  $203 

9. 

Rate  5%; 

time 

144  days; 

principal  $324 

10. 

Rate  5  %  ; 

time 

288  days; 

principal  $112 

11. 

Rate  5%; 

time 

36  days; 

principal  $408 

12. 

Rate  41%; 

time 

lyear; 

principal  $246 

13. 

Rate  4i% ; 

time 

40  days; 

principal  $444 

14. 

Rate  41%,; 

time 

80  days; 

principal  $681 

15. 

Rate  41%  ; 

time  160  days ; 

principal  $312 

16. 

Rate  6%; 

time 

2  months; 

principal  $517 

17. 

Rate  6%; 

time  4  months; 

principal  $408 

18. 

Kate  6%; 

time 

180  days; 

principal  $123 

19. 

Rate  6%; 

time 

240  days; 

principal  $204 

20. 

Rate  6%; 

time 

90  days; 

principal  $222 

157.    Six  Per  Cent  Method. 

Find  the  interest  on  $384.75  for  3  years   7   months  23 

days,  at  6%. 


Interest  85 


The  rate  for  3  years  at      The  rate  for  3  years  =  .06  x  3=     .18 
.06  per  year  is  .18.      The       The  rate  for  7  months=  .00^  x  7=    .035 

rate  for  one  month  is  j\  of      The  rate  for  23  days  =  .000^  x  23=  .003f 
.06,    or   .001,  which   gives      The  rate  for  $1  .218| 

.035     as    the    rate    for   7 

months.     The  rate  for  one  day  is  ^\  of  .00|,  or  .     $384.75 
.OOOJ-,  which  gives  .003g  as  the  rate  for  23  days.  -2181 

The  total,  .218f,  is  the  rate  for  3  years  7  months  6)1.92375 

23  days  at  .06  per  year.  .32063 

7ne  principal,  384.75,  is  then  multiplied  by  the  ^'^^^^^ 

rate  for  the  given  time,  .218|.  "'6  950 

Ahs.  $84.20  $84.19613 

The  interest  at  6%  may  be  calculated  by  multiplying  the 
principal  by  the  rate  for  the  given  time,  obtained  by  adding 
together  .06  times  the  number  of  years,  .00^  times  the  number 
of  months,  and  .OOOi  times  the  number  of  days. 

158.  Written  Exercises. 

Find  the  interest  at  6%  on  the  following; 

1.  1 1206  for  1  yr.  6  mo.  24  da. 

2.  f  375.30  for  2  yr.  8  mo.  18  da. 
•   3.    $854.10  for  1  yr.  1  mo.  6  da. 

4.  $579.60  for  3  yr.  7  mo.  12  da. 

5.  $  840  for  4  yr.  3  mo.  13  da. 

159.  Find  the  interest  on  $384.75  for  3  years  7  months 
23  days  :  (a)  at  5%  ;  (6)  at  41%  ;  (c)  at  3%  ;  {d)  at  3f  %  ; 
(e)  at  4%. 

In  Art.  157,  the  interest  on  the  foregoing  principal  for  the  given 
time  was  found  to  be  $  84.196  +. 

Interest  at  Q%=^ 84.196 
Deduct  int.  at  1  %  =     14.033  1  %  =  i  of  6  % 

(a)  Interest  at  5  %  =  $70.16  +  Ans. 

Interest  at  6%  =  $  84.196 
Deduct  int.  at  H  %  =     21.049  l-i-  %  =  i  of  6  % 

(6)  Interest  at  4i  %  =  .$63.15  -  Ans. 


86  Arithmetic 

2)$  84.196  =  Interest  at  6%. 

(c)  Interest  at  37o=  42.10  -   Ans. 

Interest  at  6  %  =  184.196 
j  Interest  at  3  %  =  ^  42.098  3  %  =  ^  of  6  % 

I  Interest  at  |  %  =     10.524  |  "/o  =  i  of  3  % 

(d)  Interest  at  3|  %  =  $  52.62  +  Ans. 

Interest  at  6  %  =  $  84. 196 
Deduct  interest  at  2  %  =     28.065  2  %  =  i  of  6  % 

(e)  Interest  at  4  %  =  $  56.13  +  Ans. 

160.  Find  the  interest  on  $  354.60  for  3  years  9  months 
17  days.     Find  the  amount  at  each  rate. 

1.  At  6    %        3.    At  7    %  5.    At  5    %         7.   At  41  % 

2.  At3|-%        4.    At3f%         6.    At  4^%  8.   At  5^  % 

Note.  The  pupil  will  observe  that  while  the  interest  at  3  %  is  one 
half  the  interest  at  6%,  the  same  is  not  true  as  to  the  respective 
anfiounts. 

Find  the  amount : 
9.   Principal  ^  1742;  rate  4%  ;  time  1  yr.  3  mo.  20  da. 

10.  Principal  1 1260 ;  rate  7%  ;  time  2  yr.  4  mo.  6  da. 

11.  Principal  $484;  rate  5%  ;  time  3  yr.  6  mo.  9  da. 

12.  Principal  $  216;  rate  3^%  ;  time  4  yr.  10.  mo.  12  da. 

13.  Principal  $  350;  rate  41%  ;  time  5  yr.  3  mo.  5  da. 

EXACT  INTEREST. 

161.  In  all  government  transactions,  interest  is  calculated 
on  the  basis  of  a  year  of  365  days.  This  is  called  exact 
interest. 

Business  banks,  in  computing  interest  due  them,  employ  the  year 
of  360  days,  but  they  pay  out  interest  on  the  basis  of  the  365-day  year. 

162.  Written  Exercises. 
Find  the  exact  interest : 

1.    On  $346.50  for  187  days  at  6%. 

$346.50  XxBoX  Ml- 


^  Interest  87 

2.  On  $  73  for  92  days  at  4%. 

3.  On  $147  at  5%  from  March  1  to  May  13. 

4.  On  $284  at  3.65%  from  June  12  to  Sept.  3. 

5.  On  1109.50  at  4i%  for  27  days. 

INTEREST-BEARING   NOTES. 

163.  A  promissory  note  is  a  written  promise  to  pay  a 
specified  sum  of  money.  The  person  signing  the  note  is  the 
maker  ;  the  payee  is  the  person  to  whom  the  note  is  made 
payable. 

164.  1. 


~''/<ff<^^^i  v/v//^//  -^^x^^^    -A-^-nSCtAjt/lti^     ajf'    -^<?-<^  xi6t^    C&^t^ 


Find  the  amount  of  the  above  note  at  maturity,  Nov.  16, 
305  days. 

2.  '  Marion,  Ind.,  Dec.  27,  1907. 
On  demand  after  date  I  promise  to  pay  to  Clement  Marsh, 

or  order,  Three  Hundred  Seventy -five  -^-^-^  Dollars,  value  re- 
ceived, with  interest  at  five  per  cent. 

$375yQoQq  Louis  Haff. 

How  much  was  due  on  this  note  at  settlement,  April  4, 
1908? 

3.  A  note  for  $360,  dated  July  20,  1907,  was  paid  with 
interest  March  18,  1908.     Find  the  amount. 


88  Arithmetic 

4.  What  sum  will  pay  a  note  for  $1080  drawn  Sept.  25, 
and  due  Dec.  31,  with  interest  at  5^%? 

5.  Find  the  amount  of  a  note  for  $760  dated  Oct.  29, 
1907,  and  paid  Feb.  29,  1908,  with  interest  at  6  per  cent. 

PARTIAL   PAYMENTS. 

165.    Merchants'  Kule. 

Grand  Rapids,  Wis.,  Oct.  10,  1907. 

On  demand  after  date  I  promise  to  pay  to  the  order  of 
Thomas  May  Two  Hundred  Seventy  Dollars,  value  received, 
with  interest  at  5  per  cent. 

$270yVo  '  Nathan  S.  Johns. 

On  this  note  the  following  payments  were  made : 

Nov.  16,  1907  ;  $  50. 
Dec.  30,  1907 ;  $  100. 
Jan.  18,  1908  ;  $  75. 

Find  the  balance  due  at  settlement,  March  29,  1908. 

Face  of  note  $  270. 

Interest  on  $270  Oct.  10  to  March  29,  171  days  6.41 

Amount  •$  276.41 

Payment  Nov.  16,  1907  $50. 

Interest  Nov.  17  to  March  29,  134  days  .93 

Payment  Dec.  30,  1907  100. 

Interest  Dec.  30  to  Mar.  29,  90  days  1.25 

Payment  Jan.  18,  1908  75. 

Interest  Jan.  18  to  Mar.  29,  71  days  .74        227.92 

Balance  due  $48.49 

When  partial  payments  are  made  on  an  interest-bearing 
note,  the  sum  due  at  settlement,  if  made  within  a  year  after 
the  note  is  drawn,  is  found  by  the  following  method : 

Find  the  amount  of  the  note  to  the  time  of  settlement,  and 
from  it  deduct  the  sum  of  the  amounts  of  the  payments,  the 
interest  on  each  of  the  latter  being  calculated  from  the  date, 
of  each  payment  to  the  date  of  settlement. 


Interest  89 

Note.  When  a  partial  payment  is  made,  it  is  generally  written 
on  the  back  of  a  note.  For  this  reason  the  payment  is  said  to  be 
indorsed. 

166.  Find  the  sum  due  at  settlement  of  the  following 
notes: 

1.  Drawn  Jan.  9,  1907;  face  $500;  rate  5%;  settled 
Dec.  18,  1907.     Payment  of  .^  300,  July  16,  1907. 

2.  Drawn  June  15,  1909  ;  face  $  1000 ;  rate  6  %  ;  settled 
Jan.  7,  1910.     Payments  : 

Aug.  12,  1910;  $300. 
Dec.  15,  1910  ;  $  500. 

3.  Drawn  Jan.  8,  1908 ;  face  $  600  ;  rate  6  %  ;  settled 
Dec.  8,  1908.     Payments  : 

April  8,  1908 ;  $  200. 
July  8,  1908;  $200. 
Oct.    8,  1908;  $200. 

4.  Drawn  July  5, 1909;  face  $1500;  rate  5%;  settled 
June  16,  1910.     Payments  : 

Sept.  8,  1909;  $200. 
Nov.  3,  1909  ;  $  300. 
Feb.  10,  1910;  $400. 
Apr.  9,  1910  ;  $  500. 

167.  United  States  Rule. 

Flint,  Mich.,  Nov.  5,  1906. 

On  demand,  I  promise  to  pay  to  George  J.  Gilmartin,  or 
order.  Eight  Hundred  Seventy  -f-^  Dollars,  value  received, 
with  interest  at  six  per  cent. 

Samuel  B.  Donaldson. 

$870JLV 

On  this  note  are  indorsed  the  following  payments ;  Dec. 
5,  1907,  $50;  May  4,  1909,  $300;  Dec.  6,  1909,  $300. 
What  sum  is  due  at  settlement,  Nov.  5,  1910  ? 


90 


Arithmetic 


Face  of  note   •...♦.. 
Interest  on  $870  from  Nov.  5,  1906,  to  Dec.  5, 

1907,  1  yr.  1  mo 

As  the  payment  of  $  50  is  less  than  the  interest, 
the  interest  on  1 870  is  computed  from  Dec. 
5,  1907,  to  May  4,  1909,  1  yr.  4  mo.  29  da. 

Amount  May  4,  1909  .... 

Less  payment  Dec.  5,  1907      .... 
Less  payment  May  4,  1909     .... 

Due  May  4,  1909 

Interest  from  May  4,  1909,  to  Dec.  6,  1909,  216  da 

Amount  Dec.  6,  1909 

Payment  Dec.  6,  1909 

Due  Dec.  6,  1909 

Interest  from  Dec.  6,  1909,  to  Nov.  5,  1910,  334  da 

Due  Nov.  5,  1910 


!  50 
300 


$870. 
56.55 

7-3.805 
$1000.355 

350. 


$  650.355 

23.412 

$673,767 

300. 

$373,767 

20.806 

$394,573 

Ans.  $  394.57 


Fiyid  the  amount  of  the  principal  to  the  time  ivhen  a  pay- 
ment, or  the  sum  of  tivo  or  more  payments,  is  at  least  equal  to 
the  interest  then  due.  From  this  amount  subtract  the  payment 
or  payments. 

Consider  the  remainder  as  a  new  principal  and  proceed  as 
before. 

168.  As  there  is  no  requirement  in  the  foregoing  note  that  interest 
is  due  at  any  specified  time,  it  would  be  unfair  to  the  maker  of  the 
note  to  add  $56.35  interest  and  deduct  the  first  payment  of  $50,  thus 
requiring  him  to  pay  interest  for  a  time  on  $876.35,  a  sum  greater 
than  the  face  of  the  note.  The  rule  that  interest  shall  not  be  required 
on  interest  prevails  in  most  of  the  states. 

169.  In  these  and  all  other  examples  in  interest,  the  time  exceed- 
ing a  year  is  found  in  years,  months,  and  days  ;  time  less  than  a  year 
is  found  by  taking  the  exact  number  of  days.  The  year  is  considered 
as  having  360  days,  except  in  the  examples  specifying  exact  interest. 

170.  Find  the  amount  due  at  settlement : 

1.  Face  of  note  $200;  rate  6%;  drawn  Jan.  15,  1908; 
date  of  settlement  July  20,  1910.  Payment  $  100  May  5, 
1909. 


Compound  Interest  91 

2.  Face  of  note  |  600 ;  rate  6%  ;  drawn  April  6,  1906; 
date  of  settlement  Nov.  1,  1910.  Payments,  $  50  Jan.  7, 
1908  ;  $  250  May  1,  1909. 

3.  Pace  of  note  $800;  rate  6%;  drawn  Aug.  21,  1906; 
date  of  settlement  Aug.  21,  1910.  Payments,  $  200  Sept. 
21,  1907  ;  $  200  Oct.  21,  1908 ;  $  200  Nov.  24,  1909. 

4.  Pace  of  note  $1200;  rate  6%  ;  drawn  Nov.  13,  1905; 
settled  Nov.  13,  1909.  Payments,  $60  Nov.  13,  1906; 
$  500  April  19,  1907 ;  $  50  May  23,  1907 ;  $  500  June  18, 
1908. 

COMPOUND   INTEREST. 

171.  A  deposit  of  $  100  is  made  in  a  savings  bank  July  1, 
1907.  On  Jan.  1,  1908,  the  depositor  is  entitled  to  ^  year's 
interest  at  4%  per  annum,  or  $2.  If  he  does  not  withdraw 
the  interest,  it  is  credited  to  his  account,  and  July  1,  1908, 
he  is  entitled  to  interest  on  $  102,  or  $  2.04.  If  this  is  not 
withdrawn,  he  is  entitled  on  Jan.  1,  1909,  to  ^  year's 
interest  on  $  104.04,  which  is  $  2.08. 

172.  Compound  interest  is  interest  taken  at  regular  periods 
upon  the  principal  and  unpaid  interest. 

173.  Written  Exercises. 

1.    Pind   the  compound   interest   on   $600    for   3   years 

6  months  at  6%.    * 

Principal,  3  600.00 

Interest,  1st  year,  36.00 

Amount,  1st  year,  8  636.00 

Interest,  2d  year,  38.16 

Amount,  2d  year,  8  674.16 

Interest,  3d  year.  40.4496 

Amount,  3d  year,  8  714.6096 

Interest,  6  months,  21.4383 

Amount,  3  yr.  6  mo.,  8  736.05 

Principal,  600.00 

Compound  Interest,  3  yr.  6  mo.,  8  136.05  Ans. 


92  Arithmetic 

2.  Find  tlie  interest  on  $  600  for  3  years  6  months  at 
6%,  compounded  semi-annually. 

Principal,  $600.00 

Interest  6  montlis,        18.00 

Amount  6  months,  ^  618.00 

Interest  6  months,        18.54 

Amount,  1  year,  $  636.54 

etc.  etc. 

To  find  the  compound  interest,  calculate  the  amount  for  the 
first  period.  Using  this  as  a  new  principal,  calculate  the 
amount  for  the  next  2ieriod,  coyitinuing  in  this  tvay  until 
the  end  of  the  last  period.  The  difference  between  the  last 
amount  and  the  principal  will  he  the  compound  interest. 

3.  Find  the  amount  of  $800  for  4  years  at  5%,  com- 
pound interest. 

4.  What  is  the  difference  between  the  simple  interest 
of  f  1000  at  4%  for  3  years  and  interest  at  the  same  rate 
compounded  quarterly  ? 

5.  Find  the  amount  of  $800  for  4  years  at  5%  com- 
pounded semi-annually. 

Principal,  $800.00 
Interest  for  6  mo.  J^       20.00 

Divide  by  4,  writing  the  first  quotient  figure  one  place  to  the  right. 

6.  How  much  less  will  be  the  interest  on  $  800  for  4  years 
at  5%,  when  compounded  semi-annually  than  when  com- 
pounded quarterly  ? 

ANNUAL  INTEREST. 

174.  Alpena,  Mich.,  July  1,  1906. 

On  demand  after  date  I  promise  to  pay  to  the  order  of 
Thomas  Tierney,  Four  Hundred  Dollars,  value  received, 
with  interest  at  six  per  cent  payable  annually. 

$400y^Q0g^  John  J.  Baknicle. 


Annual   Interest  93 

By  the  terms  of  this  note  an  interest  payment  of  $  24 
was  due  July  1,  1907.  In  the  case  of  non-payment  at  that 
date,  the  laws  of  some  states  permit  the  collection  of  simple 
interest  on  the  interest  due  until  it  is  paid. 

175.  Annual  interest  is  simple  interest  on  the  principal 
and  upon  each  deferred  interest  payment  for  the  time  each 
remains  unpaid. 

176.  Find  the   amount   due   on   the   foregoing  note   on 

July  1,  1911,  by  annual  interest,  provided  no  payments  are 

made. 

Principal,  -8  400. 
Interest  July  1,  1906  to  July  1,  1911  ;  5  years  at  6%     120. 


Interest  July  1,  1907,  to  July  1,  1911 
Interest  July  1,  1908,  to  July  1,  1911 
Interest  July  1,  1909,  to  July  1,  1911 
Interest  July  1,  1910,  to  July  1,  1911 


4  years  at  6%  on  624        5.76 

3  years  at  6%  on  6  24 

2  years  at  6%  on  6  24 

1  year  at  6%  on  6  24  


Amount  due  July  1,  1911,  8 

It  will  be  noted  that  there  are  several  items  of  interest  on  $  24, 
making  an  aggregate  of  10  years'  interest  on  $24. 

To  find  the  amount  due  on  a  note  bearing  annual  interest, 
add  to  the  amount  of  the  2)rincipal  at  simple  interest,  the 
simple  interest  on  each  unpaid  annual  interest  for  the  time  it 
remained  unpaid. 

111.   Written  Exercises. 

1.  What  is  the  difference  between  the  simple  interest 
on  $400  at  6%  for  5  years,  and  the  compound  interest? 
Between  the  simple  interest  and  the  annual  interest  ?  Be- 
tween the  annual  interest  and  the  compound  interest  ? 

2.  Find  the  annual  interest  on  a  note  for  $  1000  at  5% 
in  6  years,  no  payment  of  any  kind  having  been  made. 

3.  What  is  due  after  6  years  on  a  note  for  $  1000  bear- 
ing 6%  annual  interest,  the  first  three  interest  payments 
having  been  made  ? 


94  Arithmetic 

4.  Eobert  P.  Webb  gives  a  note  for  $  1000  dated  July 
1,  1906,  agreeing  to  pay  annual  interest  at  6%.  He  pays 
$  500  July  1, 1908.     Find  the  amount  due  July  1, 1910. 

PRESENT  WORTH. 

178.  If  I  owe  a  debt  of  $  106  payable  in  a  year,  and 
money  is  worth  6  %,  a  present  payment  of  $  100  should 
cancel  the  indebtedness,  since  the  latter  sum  will  amount  in 
a  year,  at  6%,  to  ^  106. 

179.  The  present  worth  of  $  106  payable  in  a  year  is  $  100 
when  the  prevailing  rate  of  interest  is  6%.  The  true  dis- 
count in  this  case  is  $  106  —  $  100,  or  $  6. 

180.  The  present  worth  of  a  given  sum  is  the  sum  that, 
if  placed  at  interest  at  the  given  rate,  will  amount  to  the 
former  sum  at  the  time  the  former  sum  is  payable. 

181.  To  find  the  present  worth  of  a  given  sum,  payable  at 
a  future  date,  divide  the  given  sum  by  the  amount  of  one  dollar 
for  the  given  time  at  the  given  rate. 

182.  To  find  the  true  discount,  subtract  the  present  icorth 
from  the  sum  payable  at  the  future  date. 

183.  Problems  in  present  worth  are  interesting  rather  than  prac- 
tical. Persons  to  whom  money  is  payable  at  a  future  date,  as  a  rule, 
agree  to  accept  the  sum  at  once  less  a  discount  of  a  given  per  cent, 
which  is  frequently  much  larger  than  the  prevailing  rate  of  interest. 

,184.    Written  Exercises. 

1.  What  sum  placed  at  interest  for  9  months  at  6%  will 
amount  to  1376.20? 

2.  Find  the  true  discount  on  $376.20  paid  9  months 
before  it  is  due,  money  being  worth  6%. 


Banks  and   Banking 


95 


Deposited  in 
THE  UNION  NATIONAL  BANK 


3.  What  is  the  bank  discount  of  $376.20  for  9  months  at 
6%? 

4.  I  owe  ^2500  payable  in  1  yr.  6  mo.     What  sum  in- 
vested at  4%  will  amount  to  that  sum  when  it  is  due? 

5.  Find   the    present   worth    and    the   true    discount   of 
$2500  due  in  2  years,  money  being  worth  6%. 

BANKS   AND   BANKING. 

185.  Banks  are  of  two  general  kinds,  which  may  be  dis- 
tinguished as  bus- 
iness   banks    and 
savings  banks. 

186.  The  ordi- 
nary business 
bank,  or  bank  of 
discount  and  de- 
posit, is  a  stock 
company  organ- 
ized according  to 
law  with  a  paid- 
up  capital  not  less 
than  a  certain 
specified  sum. 
The  bank  is  re- 
quired to  keep  on 
hand  at  all  times 
a  certain  percent- 
age of  the  total 
sum  due  deposit- 
ors to  meet  their 
demands. 


Racine,  Wis.,    %/mQJI    lOO'/T 


Bills 
Specie 
Checks,  enter 


separately 


DOLLARS 

550 

(0/0 

lU 


oas- 


CENTS 


75 

80 


55 


187.  A  person  depositing  money  fills  out  a  deposit  slip  in 
the  accompanying  form  and  receives  a  bank  book  in  which 
the  deposit  is  entered. 


96 


Arithmetic 


188.  The  depositor's  book  is  ruled  as  shown  in  the 
accompanying  illustration.  On  the  left  are  entered  the 
deposits  as  they  are  made ;  on  the  opposite  page  are  entered 
in  double  columns  the  sums  paid  to  tiie  depositor  or  to 
others  on  his  written  order. 


THE  UNION  NATIONAL  BANK 

of  RACINE,  WISCONSIN, 

,                                      In  account  with 

Dr. 

9(ylvn  dJO^ 

Cr. 

iqos 

t)j^. 

■    1 
1 

028 

55 

lb 

^3 

55 

10 

%ujnsJI 

120 

189.  A  depositor  withdraws  money  from  his  bank  by 
means  of  a  check.  The  illustration  on  the  opposite  page 
shows  one  form  of  a  check  book ;  the  portion  on  the  right, 
constituting  the  check,  is  detached;  and  the  other,  called 
the  stub,  is  retained  in  the  book  as  a  memorandum  for  the 
depositor  of  the  amounts  withdrawn. 

190.  The  average  business  bank  pays  no  interest  on 
moneys  deposited,  except  by  special  arrangement.  The  bank 
takes  care  of  the  money  of  its  depositors,  enables  them  to 
settle  their  bills  by  sending  the  amounts  due  through  the 
mails  in  the  form  of  checks,  and  loans  money  to  them  on 
approved  security. 

191.  A  savings  bank  is  organized  to  tA,ke  charge  of  small 
sums.  To  encourage  thrift  in  the  community,  small  de- 
posits   are    solicited,    sums    of    one    dollar,    or   less,   being 


Bank   Discount 


97 


received.  Interest  is  paid 
or  credited  to  the  deposi- 
tors at  stated  periods. 

192.  A  bank  book  is 
given  the  depositor,  as  in 
the  case  of  the  business 
bank.  Withdrawals  of 
money  are  not  made  by 
check;  the  depositor,  as 'a 
rule,  presents  himself  with 
his  bank  book,  in  which 
the  sum  withdrawn  is  en- 
tered at  the  time.  The 
interest  to  which  the  de- 
positor is  entitled  is  also 
entered  in  his  bank   book. 

BANK   DISCOUNT. 

193.  In  addition  to  loan- 
ing money  on  real  estate  or 
other  security,  banks  lend 
money  for  short  periods  to 
responsible  customers  on 
the  personal  security  of 
the  borrower,  the  repay, 
ment  of  the  money  when 
due  being,  as  a  rule,  guar, 
anteed  by  another  reliable 
person. 

194.  John  A.  Wilder  de- 
sires to  borrow  $  1500  from 
his  bank  for  90  days. 
Frank  H.  Hartridge  is  wil- 


98  Arithmetic 

ling  to  become  security  for  the  repayment  of  the  loan.     Mr. 
Wilder  makes  out  his  note  in  the  following  form : 


%J^OOf^         'PkLlaaclpkia;Pa.:^4^'fl9i^ 

s.y_J'jC^r\X^AY    CM»-A>y42  _ after  d. ale  ^^proTiitse 

to  pav  io  tke  order  of  _     A^i^^^-^-y^-ZC^j.^"/,  t^/ti^i^:^^ 

<:TA.A-^tjU^^r^\7^^-^'*'^^^^  *— — ^.  Do  Liars 

at  X^  ^  >^^AA^_/3„«^:h.  Kl,       y)cXviC  received. 


Frank  H.  Hartridge  indorses  the  note  by  writing  his  name 
on  the  back,  thereby  agreeing  to  pay  the  face  of  the  note 
when  due,  in  case  Mr.  Wilder  fails  to  meet  his  obligation. 
Mr.  Hartridge's  indorsement  also  transfers  the  note  to  the 
bearer,  the  Deshler  National  Bank,  if  the  bank  authorities 
advance  the  money. 

The  bank  discounts  the  note  by  handing  over  to  Mr. 
Wilder,  or  placing  to  his  credit,  $1477.50;  that  is,  $1500 
less  I  22.50,  for  90  days'  interest. 

195.  The  face  of  the  note  is  $  1500 ;  the  interest  deducted 
in  advance,  f  22.50,  is  called  the  bank  discount;  the  balance 
paid  or  credited  to  Mr.  Wilder,  $  1477.50,  is  called  the  j^^'o- 
ceeds.  The  note  is  due  90  days  after  Dec.  4, 1907,  or  March 
3,  1908,  which  is  called  the  day  of  maturity.  The  number 
of  days  from  the  time  of  discounting  the  note  to  the  day  of 
maturity  is  called  the  term  of  discount. 

196.  In  a  few  states  the  law  grants  the  maker  of  a  note 
three  days,  called  days  of  grace,  in  addition  to  the  time 
specified  in  the  note.  Interest  is  charged  for  these  days, 
as  well  as  for  any  other  days  allowed  by  law.      In  some 


Bank  Discount  99 

states  and.  cities  in  which  Saturday  is  a  half  holiday,  a  note 
regularly  falling  due  on  Saturday  is  not  payable  until  Mon- 
day ;  in  others,  a  note  falling  due  on  a  holiday  is  payable  the 
next  previous  business  day. 

Note.  In  the  examples  in  bank  discount  no  notice  will  be  taken  of 
holidays  or  days  of  grace. 

197-  In  discounting  a  note,  the  bank  deducts  interest  for  the 
period  between  the  date  of  discount  and  the  date  of  maturity.  Notes 
intended  for  discount  are  generally  presented  the  day  they  are  drawn, 
but  in  some  cases  the  holder  of  the  note  does  not  present  it  for  dis- 
count at  once.  In  this  case  the  term  of  discount  is  shorter  than  the 
time  for  which  the  note  is  drawn. 

198.  Notes  offered  for  discount  are  generally  drawn  for 
30,  60,  or  90  days ;  or  for  1,  2,  3,  or  4  months.  A  30-days 
note  drawn  Feb.  1,  1909,  is  due  30  days  thereafter,  or  March. 
3 ;  a  1-month  note  is  due  March  1. 

*   Preliminary  Exercises. 

199.  Find  the  date  of  maturity  of  the  following  notes : 

1.  30-days  note  drawn  Jan.  15. 

2.  1-month  note  drawn  Jan.  15. 

3.  60-day s  note  drawn  Feb.  1. 

4.  2-months  note  drawn  Feb.  1. 

5.  90-days  note  drawn  June  1. 

6.  3-months  note  drawn  June  1. 

200.  Find  the  term  of  discount: 

1.  Discounted  Feb.  1,  due  March  1. 

2.  Discounted  March  3,  due  April  15. 

3.  Discounted  Feb.  29,  due  May  5. 

4.  Discounted  Nov.  1,  due  Dec.  30. 

5.  Discounted  June  25,  due  Aug.  4. 

6.  Discounted  Sept.  1,  due  Nov.  1, 


lOO 


Arithmetic 


•  TO  FIND   THE   BANK   DISCOUNT. 

Oral  Exercises. 

201.    Find  the  bank  discount: 

Note.     The  bank  discount  is  the  interest  for  the  term.     When  no 
rate  is  specified,  6%  is  understood. 


1.  Term  60  days;  face  f  475. 
face  $300. 
face  $100. 
face  $  160. 
face  $  600. 
face  $  240. 


2.  Term  40  days 

3.  Term  90  days 

4.  Term  15  days 

5.  Term  32  days 

6.  Term  18  days 


202.    Find  the  proceeds  : 

Note.     To  find  the  proceeds,  deduct  the  discount  from  the  face. 


1.  Face  $475 

2.  Face  $  300 

3.  Face  $  100 

4.  Face  $  160 

5.  Face  $  600 

6.  Face  $  240 


term  60  days, 
term  40  days, 
term  90  days, 
term  15  days, 
term  32  days, 
term  18  days. 


203.  Find   the   bank    discount    on    a    note   for    $  600 : 

1.  Due  March  1,  discounted  Feb.  2. 

2.  Due  April  15,  discounted  March  3. 

3.  Due  May  5,  discounted  Feb.  29. 

4.  Due  Dec.  30,  discounted  Nov.  1. 

5.  Due  Aug.  4,  discounted  June  25. 

6.  Due  Nov.  1,  discounted  Sept.  1. 

204.  Find  the  date  of  maturity,  the  term  of  discount,  the 
bank  discount,  and  the  proceeds  : 


5      >,       5      >      > 

>  1  5       5 


Bank   Discoujtit    ■         «  ^>.  ,•       loi 

Face  $  157.80;  drawn  June  8,  1908  ;  time  3  months  ;  dis- 
counted  July  1;  rate  5|%. 

Date  of  maturity  —  3  months  after  June  8,  or  Sept.  8.     Ans. 
Term  of  discount  —  July  1  to  Sept.  8,  69  days.     -4ns. 

Interest  at  6%  on  $  157.80  for  60  days  =  $1,578 

6  days =      .1578 

3  days  =      .0789 

*  Bank  discount  at  6  %  $1.8147 

Deduct  jJ,  for  discount  at  ^%       .1512 

Bank  discount  at  5|  %  S  1.66+  Ans. 
Proceeds,  S  157.80  -  $  1.66  =  $  156^4  Ans. 

Note.  In  states  allowing  days  of  grace,  the  date  of  maturity  is 
Sept.  11,  the  term  of  discount  is  72  days,  the  bank  discount  is  8  1.74, 
and  the  proceeds  $  156.06. 

To  find  the  hank  discount  on  a  note,  find  the  interest  on  the 
face  of  the  note  for  the  term  of  discount. 

205.    Written  Exercises. 

Find  the  date  of  maturity,  the  term  of  discount,  the  bank 
discount,  and  the  proceeds : 

1.  Face  $346.50;   drawn  June  8,  1909;   time  90  days; 

discounted  July  1,  1909;  rate  6%. 

Date  of  maturity,  90  days  after  June  8  —  Sept.  6; 
Term  of  discount  —  67  days. 

2.  Face  $  540 ;  drawn  Feb.  4,  1908 ;  time  30  days ;  dis- 
counted Feb.  4;  rate  6%. 

3.  Face  $1224;  drawn  Jan.  6,  1908;  time  60  days; 
discounted  Jan.  16,  1908;^  rate  6%. 

4.  Face  $874.50;  drawn  Nov.  30,  1909;  time  3  months; 
discounted  Nov.  30,  1909;  rate  6%. 

Due  Feb.  28,  1910. 

5.  Face  $376.20;  drawn  Aug.  4,  1907;  time  30  days; 
discounted  Aug.  14,  1907;  rate  5%. 


IQ2  Arithmetic 

6.  Face  $4000;    drawn   Feb.    29,  1908;   time  90  days;, 
discounted  March  30,  1908;  rate  51%. 

7.  Face  $573.30;  drawn  Dec.  8.  1910;  time  4  months; 
discounted  Dec.  8,  1910;  rate  6%. 

8.  Face  $1872 ;    drawn  March  9,  1909 ;    time  60    days  ; 
discounted  April  29,  1909;  rate  4^%. 

9.  Face  $  351.90  ;  drawn  Oct.  20, 1908  ;  time  2  months  ; 
discounted  Oct.  20,  1908 ;  rate  6%.    ' 

DISCOUNT   OF   INTEREST-BEARING   NOTES. 

206.    In  settlement  of  his  account  Mr.  Niver  gives  the 
following  note  to  Sallivan,  McDonald,  &  Co. 


Buffalo,   N.Y.,   ^//(ciy  If^   190(?, 
S^aui  TyLantkoy  after  date  c/  promise  to  pay  to  the  order 

€icfktu-iawv    j~.    Dollars,  value  received,  at  the  Marine 
Bank,  with  interest  at  six  per  cent. 


This  note  is  discounted  at  6%  by  the  Marine  Bank  the 
day  it  is  drawn.     Find  the  proceeds.  * 

Date  of  maturity  =  May  4  +  4  months,  or  Sept.  4 
Term  (27  +30  +  31  +  31+4)  days,  or  123  days 

Face  of  note  —  $  784. 75 

{Interest  for  60  days  7.8o- 

Interest  for  60  days  7.85- 

Interest  for    3  days  .39 

Amount  due  Sept.  4  .'$800.84 


Discount  of  Interest-Bearing  Notes        103 


{Discount  for  60  days 
Discount  for  60  days 
Discount  for    3  days 


Proceeds,  May  4  $784.42 

Note.  It  will  be  observed  that  the  bank  discount,  $  16.42,  exceeds 
the  interest,  $  16.09,  by  33  cents.  This  difference  of  33  cents  is  the 
interest  for  123  days  on  §  16.09,  the  interest  on  the  face  of  the  note. 

To  find  the  hai}k  discount  on  an  interest-bearing  note,  find, 
the  interest  for  the  term  of  discount  on  the  amount  of  the  note 
at  maturity. 

207.  In  a  non-interest-bearing  note,  the  discount  is  taken 
on  the  sum  given  in  the  face  of  the  note.  In  an  interest- 
bearing  note,  the  sum  due  at  maturity  is  the  amount  of  the 
note ;  that  is,  the  face  of  the  note  plus  the  interest  for  the 
time  stated  in  the  note. 

208.  Written  Exercises. 

Find  the  proceeds  of  the  following  interest-bearing  notes, 
the  rate  of  interest  in  each  case  being  6%,  and  the  rate  of 
discount  6% : 

1.  Sixty-days  note  for  $1872,  drawn  March  9,  1909, 
discounted  April  29,  1909. 

2.  Ninety -days  note  for  $4000,  drawn  Feb.  29,  1908, 
discounted  March  30,  1908. 

3.  Thirty-days  note  for  $376.20,  drawn  Aug.  4,  1907, 
discounted  Aug.  14,  1907. 

4.  Sixty-days  note  for  $1224,  drawn  Jan.  6,  1908,  dis- 
counted Jan.  16,  1908. 

5.  Three-months  note  for  $157.80,  drawn  June  8,  1908, 
discounted  July  1,  1908. 


I04  Arithmetic 

PROBLEMS   IN   INTEREST   AND   BANK   DISCOUNT. 
TO   FIND   THE   TIME. 

209.  Preliminary  Exercises. 

1.  Find  the  interest  on  $  200  for  3  years  at  5%. 

2.  The  interest  on  $  200  at  5%  is  $  30.     Find  the  time. 

3.  The  interest  on  f  200  for  3  years  is  $  30.     Find  the 
rate. 

4.  The  interest  for  3  years  at  5%   is  $30.     Find  the 
principal. 

5.  Interest   $36;    time   3   years;    rate  6%.      Find   the 
principal. 

Written  Exercises. 

210.  Find  the  time  in  which  a  principal  of  $  240  at  6  % 
will  yield  interest  as  follows  : 

1.  114.40  4.   $18  7.    $1.80 

2.  $    1.44  5.    $19.80  8.    $0.72 

3.  $15.84  6.    $21.60  9.    $0.04 

211.  Find  the  time  in  which  a  principal  of  $  240  at  6% 
will  produce  amounts  as  follows : 

1.  $254.40  4.    $259  7.    $245.76 

2.  $241.44  5.    $260.80  8.    $240.68 

3.  $255.84  6.   $262.60  9.    $240.08 

212.  In  what  time  will   $300  at  6%   yield  $49.95  in- 
terest ? 

Representing  time  by  T,  we  have 

300  X  if  0  X  T  =  49.95 
or  18  r=  49.95 

T  =  49.95  -^  18 


Problems  In   Interest  and    Bank    Discount      105 

The  time  in  years  is  j^g  of  49.95.  Dividing  49.95  years  by  18  gives  a 
quotient  of  2  years  and  a  remainder  of  13.95  years.  This  is  reduced  to 
167.4  months.  Dividing  by  18,  gives  a  quotient  of  9  months  and  a  re- 
mainder of  5.4  months,  or  162  days.  Dividing  162  days  by  18  gives 
a  quotient  of  9  days. 

18)49.95  years(2  years 
36. 
remainder      13.95  years 

or  13.95  times  12  months 


18)167.4  months(9  months 
162 
remainder  5.4  months 

or  5.4  times  30  days 

18)T62jO  days(9  days 
162 

Ans.  2  yr.  9  mo.  9  da. 

An  examination  of  the  foregoing  will  show  that  the  time  in  years  is 
found  by  dividing  the  interest,  $49.95,  by  the  interest  on  '$300  at  6% 
for  1  year,  which  is  318. 

That  is,  if  •$  18  interest  is  produced  in  a  year,  the  number  of  years 
required  to  produce  $49.55  interest  will  be  the  quotient  of  49.55  by  18. 

To  find  the  time  in  years,  divide  the  given  interest  by  the 
interest  for  one  year  on  the  given  princijml  at  the  given  rate. 

213-  When  the  amount  is  given,  the  interest  is  found  by  deducting 
the  principal  from  the  amount. 

The  foregoing  problem  might  read  :  In  what  time  will  $300  at  6% 
amount  to  $349.95? 

In  this  case  the  interest  is  found  to  be  $349.95  —  $300,  or  $49.95, 
after  which  the  problem  is  solved  as  shown  above. 

Proof  : 

Principal  $300 

6%  18.00  interest  for  1  year 

6%  18.00  interest  for  1  year 

i  year  9.00  ifiterest  for  6  months 

^  of  6  months  4.50  interest  for  3  months 
^  of  3  months  .45  interest  for  9  days 


$349.95  amount  for  2  yr.  9  mo.  9  da. 


To6  Arithmetic 

214.  Find  the  time : 

1.  Principal  f  240  ;  rate  6%  ;  interest  ^  14.40. 

2.  Principal  $250;  rate  5%  ;  amount  $262.50. 

3.  Principal  $300;  rate  4%  ;  interest  $14.50. 

4.  Principal  $300;  rate  4%  ;  amount  $315.50. 

5.  Principal  $800  ;  rate  4i%  ;  interest  $124.40. 

TO   FIND   THE   TERM    OF   DISCOUNT. 

215.  Oral  Exercises. 

1.  Principal    $300;    rate   6%;    interest   $9.      Time  in 
years  ?     In  months  ?     In  days  ? 

2.  Face    of  note  $300;    rate  6%;    bank   discount   $9. 
Term  in  months  ?     In  days  ? 

3.  Face  of  note  $300;   rate  6%;   bank  discount  $4.50. 
Term  in  months? 

4.  Face  of  note  $300;   rate  6%;    bank  discount  $0.45. 
Term  in  days  ? 

6.  Face   of   note  $600;    rate   6%;    bank   discount  $6. 
Term  in  days  ? 

Written  Exercises. 

216.  Find  the  number  of  days  in  which  a  note  of  $240, 
discounted  at  6%,  will  yield  discounts  as  follows: 

1.  $3.60  4.    $6.6S  7.    $2.92 

2.  $0.36  5.    $4.32  8.    $3.48 

3.  $3.96  6.    $1.24  9.   $1.76 

217.  Find  the  number  of  days  in  which  a  note  of  $240, 
discounted  at  6%,  will  yield  proceeds  as  follows : 

1.  $236.40  4.    $238.76  7.   $235.68 

2.  $239.64  5.   $238.24  8.   $233.32 

3.  $236.04  6.    $236.52  9.    $237.08 


Problems   in   Interest  and   Bank   Discount      107 

218.  In  what  time  will  the  bank  discount  on  a  note  for 
$300  be  ^13.95? 

As  the  term  of  discount  is  generally  required  in  days,  the  discount, 
$13.95,  is  divided  by  the  discount  (interest)  for  1  day  on  $300  at  6%. 

The  interest  on  $300  for  a  year  is  $  18  ;  for  a  day,  it  is  3!^^  of  $  18, 
or  5^.     The  time  in  days  =  13.95 -^  .05,  or  279.    ^/is.  279  days. 

To  find  the  term  of  discount  in  days,  divide  the  giveyi  dis- 
count by  the  discount  for  one  day  at  the  given  rate  on  the  given 
sum. 

219.  Find  the  term  of  discount : 

1.  Face  of  note  $240;  rate  6%  ;  bank  discount  $3.76. 

2.  Face  of  note  $840;  rate  5%  ;  proceeds  $835.10. 

3.  Face  of  note  $  1260 ;  rate  6%  ;  bank  discount  $  13.86. 

4.  Face  of  note  $1320;  rate  6%  ;  proceeds  $1311.20. 

5.  Face  of  note  $1500;  rate  5%;  bank  discount  $18.75. 

TO  FIND   THE   RATE   OF  INTEREST. 

Preliminary  Exercises. 

220.  Find  the  rate  at  which  $  200  will  yield  yearly  inter- 
est as  follows : 

1.  $10  4.    $9  7.   $6 

2.  $13  5.    $12  .  8.    $11 

3.  $7  6.   $8  9.    $14 

221.  Find  the  rate  at  which  $200  will  yield  interest  as 
follows : 


1.    $10  in  1  year. 

4.    $2  in  60  days. 

2.    $5  in  6  months. 

5.    $3  in  3  months. 

3.    $18  in  2  years. 

6.    $30  in  3  years. 

io8  Arithmetic 

222.  Find  the  rate  at  which  $  200  will  produce  the  fol- 
lowing amounts : 

1.  1210  in  1  year.  4.    ^  215  in  2  yr.  6  mo. 

2.  $228  in  2  years.  5.    $202  in  60  days. 

3.  $218  in  2  years.  6.    $  204  in  90  days. 

Written  Exercises. 

223.  Find  the  rate  at  which  $  240  will  produce  yearly 
interest  as  follows : 

1.  $12  3.   $13.20  5.   $16.80 

2.  $10.80        .  4.   $14.40  6.    $15.60 

224.  Find  the  rate  at  which  $  240  will  yield  interest  as 
follows : 

1.  $  12  in  1  year.  4.    $  2.80  in  60  days. 

2.  $6  in  6  months.  5.    $3.60  in  3  months. 

3.  $  31.20  in  2  years.  6.    $  32.40  in  3  years. 

225.  Find  the  rate  at  which  $  240  will  produce  amounts 
as  follows : 

1.  $  255.60  in  1  year.  4.    $  242.80  in  60  days. 

2.  $  244.20  in  6  months.  5.    $  243.30  in  3  months. 

3.  $  261.60  in  2  years.  6.    $  283.20  in  3  years. 

226.  At  what  rate  will  $300  yield  $49.95  interest  in 
2  years  9  months  9  days? 

Representing  the  rate  by  B,  we  have 

300  X  —  X  ^  =  49.95 
100      360 

§5§^  =  49.95 
40 

B  =  49.95  ^  — 
40 

49  95  X  40 
Inverting  the  divisor,  and  canceling,  —^ —  —  =  6.     Ans.  6  %. 


Problems  in   Interest  and   Bank   Discount      109 

An  examination  of  the  foregoing  will  show  that  the  required  rate  is 

obtained  by  dividing  the  interest,  -$49.95,  by  the  interest  on  §  300  at  1  % 

$  338 
for  the  given  time,  which  is  $ 8.325,  or  '^'       ; 

«  000 
That  is,  if  '^— —  interest  is  produced  in  the  given  time  at  1  %,  the 

rate   required  to   produce  $49.95  will  be  the  quotient  of  49.95  by 
333 
40' 

To  Jind  the  rate  of  interest,  divide  the  given  interest  by  the 
interest  at  one  ]per  cent  for  the  given  principal  for  the  given 
time. 

227.  When  the  amount  is  given,  the  interest  is  found  by  deduct- 
ing the  principal  from  the  amount.     (See  Art.  143.) 

228.  Find  the  rate  of  interest : 


1.  Principal  $  200 

2.  Principal  $  300 

3.  Principal  $  400 

4.  Principal  $  500 

5.  Principal  ^  600 


time  2  yr.  6  mo. ;  int.  $  30. 
time  60  days  ;  amt.  $  303.50. 
time  1  yr.  1  mo.  1  da. ;  int.  $  19.55. 
time  3  yr.  20  da. ;  amt.  3  568.75. 
time  4  yr.  3  mo.  6  da. ;  int.  $  166.40. 


TO   FIND  THE   RATE   OF   DISCOUNT. 

229.  The  discount  for  60  days  on  a  note  for  $  120  is  $1; 
what  is  the  rate  ? 

This  problem  resolves  itself  into  the  following  problem  in  interest : 
At  what  rate  will  9 120  yield  $  1  interest  in  (30  days  ? 

230.  When  the  proceeds  are  given,  the  discount  is  obtained  by 
deducting  the  proceeds  from  the  face  of  the  note. 

231.  Find  the  rate  of  discount : 

1.    Face    of    note     $  100 ;     term    135    days  5     discount 
S  1.50. 


iio  Arithmetic 

2.  Face     of    note    $  150 ;    term     66     days ;    proceeds 
$  148.35. 

3.  Face  of  note  $  200 ;  term  144  days ;  discount  f  2. 

4.  Face  of  note  $250;  term  72  days;    proceeds  $247. 
6.    Face  of  note  $300;  term  93  days  ;  discount  $3.10. 

TO   FIND   THE   PRINCIPAL. 

Preliminary  Exercises. 

232.  Find  the  principal  that  will  yield  yearly  interest  as 
follows  at  6%  : 

1.  $6  •  4.    $15  7.    $27 

2.  $7.20  5.    $7.50  8.    $12 

3.  $18  6.    $21  9.    $33 

233.  Find   the    principal    that   will    yield    interest    as 
follows  at  6%: 

1.  $  12  in  2  years.  4.    $  7.50  in  6  months. 

2.  $  14.40  in  2  years.  5.    $  18  in  3  years. 

3.  $  27  in  11  years.  6.   $  10  in  60  days. 

234.  Find  the  principal  that  will  produce  the  following 
amounts  at  6%  : 

1.  $  106  in  1  year.  4.    $224  in  2  years. 

2.  $  424  in  1  year.  5.    $  424  in  1  year. 

3.  $318  in  1  year.  6.    $118  in  3  years. 

"Written  Exercises. 

235.  Find  the  principal  that  will  yield  yearly  interest  as 
follows  at  6%  : 

1.  $92.34  4.    $18.36  7.   $143.04 

2.  $47.16  5.    $27.72  8.    $216.30 

3.  $33.12  6.    $51.30  9.    $403.20 


Problems  in   Interest  and   Bank   Discount      iii 

236.  Find   the   principal    that    will    yield    interest    as 
follows  at  5%  : 

1.  $  92.34  in  1  year.  4.    $18.36  in  4  years. 

2.  $47.16  in  2  years.  5.    $  27.72  in  5  years. 

3.  $33.12  in  3  years.  6.    $  51.30  in  6  years. 

237.  Find  the  principal   that  will  produce  amounts  as 

follows  at  4%  : 

1.  $  114.40  in  1  year.  '  4.    $  150.80  in  4  years. 

2.  $  162  in  2  years.  5.    $  111  in  5  years. 

3.  $  134.40  in  3  years.  6.    $  186  in  6  years. 

238.  What  principal  at  6  %  will  yield  $  49.95  interest  in 
2  years  9  months  9  days  ? 

Representing  the  principal  by  P,  we  have 
^^^xP:=  49.95 


2000 

P  -  49  95  ^  ^?i 
P_  49.95  .  2^^^ 

49  95  X  2000 
Inverting  the  divisor,  and  canceling,  — —k^k^ =  300.     Ans.  .$  300. 

ooo 

An  examination  of  the  foregoing  will  show  that  the   principal   is 

found  by  dividing  the  interest,  $  49.95,  loy  the  interest  on  §  1  for  2  years 

$  333 
9  months  9  days,  at  6%,  which  is  $  .1665,  or  '^^• 

2000 

That  is,  if  $  .1665  is  produced  in  a  given  time  by  81  principal,  the 

principal  required  to  produce  $49.95  will  be  the  quotient  of  $49.95  by 

.1665. 

To  fiyid  the  principal,  divide  the  given  interest  by  the  inter- 
est on  one  dollar  at  the  given  rate  for  the  given  time. 

239.    What  principal  at  6%  will  amount  to  $349.95  in 
2  years  9  months  9  days  ? 


112  Arithmetic 

333  P 


In  Art.  238,  the  interest : 


2000 


m.  ^   *x.      f  T,  ,  333  P     2333  P 

The  amount,  therefore,  =  P  -] = 

'       '      2000    2000 
2333  P 

Inverting  the  divisor,  and  canceling,  the  value  of  P  is  found  to  be 
300.     Ans.  .^300. 

To  Jind   the   principal,    divide    the    given   amount   by   the 
amount  of  one  dollar  at  the  given  rate  for  the  given  time, 

240.  Find  the  principal : 

1.  Interest  %  38.40 ;  time  1  yr.  6  mo. ;  rate  4%. 

2.  Amount  $  133.50  ;  time  2  yr.  3  mo.  ;  rate  5%. 

3.  Interest  $  50.40  ;  time  4  mo.  20  da. ;  rate  6%. 

4.  Amount  $  183.60 ;  time  5  mo.  10  da. ;  rate  ^%. 

5.  Interest  $49.50;  time  128  days;  rate  5i%. 

TO   FIND   THE   FACE   OF   A   NOTE. 

241.  The  discount  at  6%  for  60  days  on  a  note  is  $1; 
what  is  the  face  of  the  note  ? 

This  problem  resolves  itself  into  the  following  problem  in  interest : 
What  principal  at  6  %  will  yield  in  60  days  ^  1  interest  ? 

242.  Written  Exercises. 

Find  the  face  : 

1.  Discount  $9.84;  term  60  days ;  rate  6%. 

2.  Discount  $24.30;  term  90  days  ;  rate  6%. 

3.  Discount  $18.92;  term  120  days;  rate  6%. 

4.  Discount  $7.87;  term  &&  days;  rate  6%. 

5.  Discount  $7.32;  term  72  days;  rate  6%. 


Problems  in   Interest  and   Bank   Discount      113 

243.  The  proceeds  of  a  note  discounted  for  60  days  at  6% 

are  $  99 ;  what  is  the  face  of  the  note  ? 

Let  F  represent  face  of  note, 

Then  Fx—x  —  =  discount  =  — 
100     360  100 

Proceeds  =  F = 

100       100 

100 
Clearing  of  fractions,  99  i^=  9900 

i^=100  ^ns.  NlOO. 

To  find  the  face  of  a  note,  divide  the  given  proceeds  (discount) 
by  the  proceeds  (discount)  of  one  dollar  at  the  given  rate  for 
the  giveyi  term. 

244.  Find  the  face  : 

1.  Discount  $  15.07  ;  term  66  days  ;  rate  6%. 

2.  Proceeds  $  1354.93;  term  72  days;  rate  ^%. 

3.  Discount  ^47.50;  term  95  days;  rate  6%. 

4.  Proceeds  I?  296.35 ;  term  73  days  ;  rate  6%. 

5.  Discount  $  11 ;  term  120  days  ;  rate  51%. 


CHAPTER  IV. 

BUSINESS   FORMS   AND  USAGES;    REVIEW. 

TRANSMISSION   OF   MONEY. 

245.  If  Mr.  Calkins  in  Omaha  desires  to  send  $  25  to 
Mr.  Shaw  in  San  Francisco,  he  may  do  so  in  any  one  of 
several  ways.  He  may  send  his  check ;  a  postal  moyiey  order 
payable  in  San  Francisco  can  be  procured  in  Omaha  and 
sent  to  Mr.  Shaw  by  mail ;  an  express  company  will  sup- 
ply an  express  money  order  payable  in  San  Francisco. 

246.  Large  sums  of  money  are  generally  transmitted 
through  banks.  The  following  is  a  cashier\s  check  bought 
in  Boston  by  J.  C.  Stewart,  and  sent  to  W.  H.  Ingraham, 
Chicago,  in  settlement  of  account. 


No.  S/-^. 
CENTRAL   NATIONAL   BANK, 

BosTox,  Mass.,  ^4^^.  f3,  190<?. 
Pay  to  the  order  of  W-,  /if.  Jn(^vcLkcL^fv ^BOOO^L 

To  The  First  National  Bank,  1  €.tui.  /if.   £u/o&, 

Chicago,  111.  j  Cashier. 


247.  Mr.  Ingraham  might  collect  from  Mr.  Stewart  the 
amount  of  the  latter's  indebtedness  by  means  of  a  sight 
draft,  as  follows  ; 

114 


Business  Forms  and   Usages  1 1 5 


$2000  ^^                           Chicago,  III.,  (Lvlcj.  f3,  190^. 

At  sight  pay  to  the  order  of 

THE   CENTRAL   NATIONAL   BANK   OF  BOSTON 

c^^^   ^k(ywQ.a.ncL   ^--^^.^.^.^^^-.^-.-.x.-.-.-..-..^-^^ 

Value  received,  and  charge  to  account  of 

To  f.  ^.  ^ttw-a.\t, 

120  Bo-iftoXxyyv  eft., 

W'.  /if.  c^rupM^fiauyyiy. 

£aaZa7^,  TiicMAy. 

248.  Mr.  Ingraham  deposits  this  draft  in  his  bank  in 
Chicago,  The  First  National,  which  sends  it  for  collection 
to  The  Central  National  Bank  of  Boston.  The  Chicago 
bank  charges  Mr.  Ingraham  something  for  the  expense  of 
collection. 

249.  In  the  foregoing  draft,  W.  H.  Ingraham  is  called 
the  drawer;  J.  C.  Stewart,  the  draicee;  and  The  Central 
National  Bank,  the  payee. 

250.  The  following  is  a  form  of  draft  employed  to  collect 
a  bill  due  at  a  later  date : 


f/600  ^  Brokex  Bow,  Neb.,  f'u^n.&  /6,  \^^8. 

At  QA/?ctAi  cicbif^'  sight  pay  to  the  order  of 

THE   STOCKMEN'S   NATIONAL   BANK 
OF  FORT  BENTON,  MONT. 

Si{t£.£Av  ffwyicUe.cC  j^^^^-^^^^.-^-^^^^^-^^^-^^^^^^ Dollars. 

Value  received,  and  charge  to  account  of 
To  ^&<>uf&  ^.  ffciiyvt&t,      .  I 
ofa-ht  /o&'H'ta'yv,  Tflo-nt. 


ii6  Arithmetic 

251.  This  is  called  a  time  draft.  It  is  presented  to  Mr. 
Hamlet  on  June  19.  If  he  is  willing  to  pay  it,  he  writes  in 
red  ink  on  the  face  of  the  draft  the  word  "Accepted," 
followed  by  the  date  and  his  signature.  The  draft  then  is 
called  an  acceptance.  It  has  practically  become  a  promissory 
note  drawn  by  George  D.  Hamlet,  June  19,  1908,  payable 
60  days  thereafter  to  the  Stockmen's  National  Bank. 

252.  The  term  exchange  is  used  to  designate  transactions 
by  which  accounts  at  a  distance  are  settled  without  trans- 
mitting actual  cash.  A  bank  draft  sometimes  costs  more 
than  the  sum  named  in  the  face ;  exchange  is  then  at  a 
premium.  Under  some  circumstances  a  bank  draft  may  be 
bought  for  less  than  its  face ;  exchange  is  then  at  a  discount. 
The  newspapers  publish  the  prevailing  rates  of  exchange,  the 
discount  or  premium  given  being  the  rate  for  each  $  1000. 


BANK   DRAFTS. 

253.   Written  Exercises. 

1.  What  will  be  the  cost  of  a  draft  on  New  York  for 
$12,350,  purchased  in  Chicago  at  15^  premium  ? 

Face  of  draft,  $  12,350 
Premium,  $.15  x  12.35,  1.8525 

$12,351.8525    ^?is.  $12,351.85. 

To  find  the  cost  of  a  draft,  add  the  premium  to  the  face 
value,  or  subtract  the  discount. 

2.  Find  the  cost  of  a  draft  for  $2346.50  at  $1  discount. 

3.  Mr.  Jones  drew  on  Mr.  Cottier  through  a  bank,  for 
$4500.  How  much  did  Mr.  Jones  receive  if  the  cost  of 
collecting  the  draft  was  yo%  ?  . 

4.  The  cost  of  a  draft  for  $25,000  was  $25,015.  What 
was  the  rate  of  exchange  ? 


Foreign   Exchange  117 

5.  Mr.  Bussey  made  a  draft  on  Mr.  Mills  for  $24,000  at 
60  days'  sight.  Find  the  proceeds  of  the  draft  if  it  is  dis- 
counted at  6%  by  Mr.  Bussey's  bank,  the  term  of  discount 
being  taken  at  66  days  to  include  the  time  taken  to  send 
the  draft  to  its  destination  and  to  receive  the  money. 

FOREIGN   EXCHANGE. 

254.  The  term  foreign  bill  of  exchange  is  generally  applied 
to  bank  drafts  payable  in  a  foreign  country. 

255.  A  person  who  buys  foreign  exchange  receives  as  a 
rule  two  bills,  called  a  set.  When  either  of  the  set  is  paid, 
the  other  becomes  valueless.  In  some  cases  both  bills  are 
mailed  to  the  payee  by  different  vessels,  and  the  first  to 
reach  him  is  presented  for  payment. 

256.  The  following  two  bills  constitute  a  "set"  of  ex- 
change : 


1    Exchange  for  £1000.         Neio  York,  Sept.  1,  1908. 
At  sight  of  this  first  of  exchange,  second  unpaid,  jmy 
to  the  order  of  Joseph  Nooncin,  One  Thousand  Pounds 
Sterling,  value  received,  and  charge  to  account  of 

JOSEPH  F.  LAMORELLE  Sf  CO. 
To  Johnson  &  Chambers,  London,  Eng. 


2    Exchange  for  £1000.      New  York,  Sept.  1,  1908. 
At  sight  of  this  second  of  exchange,  first  unpaid, 
pay  to  the  order  of  Joseph  Noonan,  One  Thousand 
Pounds  Sterling,  value  received,  and  charge  to  account  of 

JOSEPH  F.    LAMORELLE   §-    CO. 
To  Johnson  &  Chambers,  London,  Eng. 


ii8  Arithmetic 

257.  A  bill  of  exchange  may  be  drawn  for  any  number 
of  days  after  sight ;  but  bankers'  time  bills  are  generally 
for  60  days. 

258.  Bills  of  exchange  differ  from  ordinary  drafts  only 
in  being  payable  in  a  foreign  currency  and  being  frequently 
made  out  in  sets  of  two  bills. 

259.  The  equivalent  values  of  the  units  of  coinage  of  the 
leading  European  countries  are  shown  in  the  following 
table : 

£1=^4.8665 
1  franc  =  f  0.193 
1  mark  =  $0,238 

The /ranc  is  the  name  of  the  unit  in  France,  Belgium,  and  Switzer- 
land. The  lira  in  Italy,  the  drachma  in  Greece,  t\\Q  peseta  in  Spain, 
and  the  mark  in  Finland  have  the  same  intrinsic  value  as  the  franc, 
19.3  cents  in  United  States  money. 

The  mark  is  the  unit  of  the  coinage  of  Germany,  and  the  pound 
sterling  that  of  England. 

For  the  value  of  the  coins  of  other  countries,  see  the  Appendix. 

260.  Rates  of  foreign  exchange  are  given  in  the  daily 
papers.     The  following  prices  were  recently  quoted: 


Sterling. 

Franc. 

Mark. 

Cable, 

4.89| 

5.14f 

.951 

Demand, 

4.88 

5.16^ 

.95| 

60-days, 

4.841 

5.19f 

.941 

261.  These  rates  show  under  the  head  of  "Sterling"  the 
cost  of  a  pound  sterling  in  United  States  dollars.  For  trans- 
fer by  cable  the  rate  is  $4.89 J  per  £.  A  sight  bill  of  ex- 
change, payable  on  presentation  in  London,  costs  $  4.88  per  £, 
while  a  bill  payable  60  days  after  presentation  costs  $4.84^ 
per  dS.  r 


Foreign   Exchange  IT9 

262.  The  quotations  under  "  francs  "  indicate  the  amount 
of  exchange  in  francs  that  can  be  bought  for  $1;  that,  for 
instance,  $  1000  will  be  the  cost  of  a  cable  transfer  of  5147.50 
francs  to  Paris;  that  the  same  sum  will  purchase  a  sight  bill 
for  5162.50  francs  or  a  60-day  bill  for  5193.75  francs. 

263.  The  quotations  under  "  marks "  give  the  cost  of 
4  marks  in  United  States  money ;  viz.,  that  a  cable  transfer 
of  4000  marks  costs  $958.75,  a  sight  bill  of  exchange  for 
the  same  sum  costs  $952.50,  and  a  60-day  bill,  $945. 

264.  EngHsh  Money. 

4  farthings  =  1  penny  (d. ) 
12  pence  =  1  shilhng  (s.) 
20  shillings  =  1  pound  (£) 

Farthings  are  generally  written  as  fractions  of  a  penny. 

French  Money. 

100  centimes  =  1  franc  (/r.) 

German  Money. 

100  pfennigs  =  1  mark  (M.) 

265.  Small  sums  are  transmitted  to  foreign  places  by 
means  of  postal  and  express  money  orders. 

266.  "Written  Exercises. 

1.  Find  the  cost  in  Boston  of  a  60-day  bill  on  Liverpool 
for  £165  13s.  Scl  at  4.85|. 

To  find  the  cost  of  the  bill  we  can  multiply  the  cost  of  1  £,  $-4.8575, 
by  the  number  of  pounds. 

Since  1  shilling  contains  12  pence,  13s.  Sd.  =  WM.  There  are  240 
pence  in  a  pound  sterling;  13s.  Scl,  therefore,  equal  if^  of  a  pound, 
or  ^1 

The  cost  of  the  bill  =  $4.8575  x  165|^. 


I20  Arithmetic 

Note.     Business  men  prefer  to  use  aliquot  parts. 


£100  = 

$485.75 

50  = 

242.875 

1  of  £  100 

10  = 

48.575 

Jo  of  £  100 

5  = 

24.2875 

J^of  £50 

10s.  = 

2.4288 

^of£5 

2s.  = 

.4858 

1  of  10s. 

ls.= 

.2429 

1  of  2s. 

8^.  = 

.1619 

1  of  2s. 

£  165  13s.  M.  =  .$  804.8069 
Ans.  §804.81. 

2.  Find  the  cost  of  £  1000  at  4.86J. 

3.  Find  the  cost  of  £  1787  10s.  at  4.85. 

4.  Mr.  Imlay  desires  to  transmit  by  cable  £375  9s.  6d. 
Find  the  cost  at  4.871. 

5.  Find  the  cost  of  a  60-day  bill  on  Paris  for  2160.75  fr. 
at  5.18f . 

At  5.18|  francs  for  $1,  the  number  of  dollars  will  be  2160.75  -f- 
5.1875. 

6.  A  merchant  owes  3000  marks  in  Berlin.  What  will 
be  the  cost  of  a  cable  transfer  at  941  ?  ^ 

The  cost  per  4  marks  being  94^  cents,  1  mark  costs  \  of  94|  cents, 
and  3000  marks  cost  (i  of  M\f)  x  3000. 

7.  I  bought  goods  in  Germany  to  the  amount  of  4520 
marks.  How  much  must  I  pay  in  New  York  for  a  bill  of 
exchange  at  95^  to  settle  the  account  ? 

8.  Find  the  cost  in  Chicago  of  a  bill  on  Hamburg  for 
10,500  marks  at  951. 

9.  When  the  rate  of  exchange  on  Berlin  is  96^,  what  is 
the  face  of  a  bill  that  costs  $  1808.41  ? 

10.  An  exporter  sold  to  a  merchant  in  Havre  goods 
amounting  to  $  3600,  and  drew  on  him  at  the  rate  of  5.20^. 
What  was  the  face  of  the  bill  in  francs  ? 


Bills  and  Accounts 


121 


11.  A  man  paid  $  1000  for  a  cable  transfer  to  Dublin  at 
4.88.     What  sum  was  paid  in  Dublin  ? 

12.  A  banker  paid  $10,000  for  a  cable  transfer  to  Paris, 
in  which  city  a  bill  of  exchange  on  London  was  purchased 
with  the  proceeds  at  25.155  francs  per  £.  What  was  the 
face  of  the  latter  bill  ? 

13.  Find  the  cost  of  a  cable  transfer  to  London  of 
£  2046  6s.  3d.  at  4.88. 


BILLS  AND  ACCOUNTS. 
BILL   FOR   GOODS   BOUGHT   AT   ONE   TIME. 

267.  Louisville,  Ky.,  Feb.  29,  1908. 

Mr.  Hunter   Collins 

Bought  of  Kelly  &  Warrex. 


25  lb.  Sugar 

1  bbl.  Flour 
10  lb.  Bacon 

2  lb.  Tea 


.051 


121 
.45 


Received  payment,  Mch.  6, 1908, 
Kelly  &  Warren, 

per  S.M.Y. 


This  is  a  bill  for  purchases  made  Feb.  29,  1908.  It  was  paid  March  6, 
1908,  to  the  clerk  who  writes  his  initials  under  the  name  of  the 
firm. 

1.  Copy  the  foregoing  bill,  filling  in  the  missing  amounts. 

2.  Make  out  a  bill  for  the  following  articles  sold  to-day 
by  M.  L.  Hutchinson  to  F.  Curtis :  12i  yd.  Dress  Goods, 
@  40^;  5  yd.  Ribbon,  @  63^;  10  yd.  Silk,  @  75^;  1  Hat, 
$  8.75.     J.  H.  Bancroft  receipts  the  bill. 


122 


Arithmetic 


BILL  FOR  GOODS  BOUGHT  AT  DIFFERENT  TIMES. 
268  Marietta,  0.,  July  1,  1908. 

W.    S.    GOODNOUGH    &    Co. 

Sold  to  L.  H.  GuLicK. 


1908 

June 

1 

5  M  Flooring 

32.00 

u 

10 

21  M  Lath 

5.00 

u 

10 

3  kegs  Nails 

2.50 

(( 

12 

4  M  Scantling 

24.00 

t( 

20 

2  M  Joists 

20.00 

(I 

20 

li  M  Scantling 

24.00 

il 

20 
30 

2  M  Lath 
By  Cash 
Balance  due 

5.15 

25 

u 

00 

a 

^ 

1.  Complete  the    foregoing   bill.     Receipt  it   for  L.  H. 
Gulick,  adding  your  initials  as  clerk. 

2.  Make  out  a  bill  of  several  items  bought  at  a  grocery 
during  the  present  month. 

BILL   FOR   SERVICES   RENDERED  AND   MATERIAL 

SUPPLIED. 

269.  Bismarck,  N.D.,  May  5,  1908. 

Mr.  James  P.  Henry 

To  A.  S.  Caswell,  Dr. 


To  5  Roses 

.25 

"   25  Geraniums 

.10 

"   3  Maples 

.75 

"   40  Pansies 

.05 

"    Labor,  Fertilizer,  etc. 

2 

75 

$ 

Bills  and  Accounts 


123 


Note.  Bills  for  goods  purchased  may  employ  the  heading 
"  Bought  of,"  "  Sold  to,"  or  the  one  used  above.  The  heading  here 
given  is  the  appropriate  one  for  a  bill  including  services  rendered. 

1.  Copy  and  complete  the  foregoing  bill. 

2.  James  H.  Tully  presents  a  bill  to  you,  dated  to-day, 
containing  the  following  items :  15  days  cutting  wood, 
@  $  1.75 ;  6  days  hauling  ice,  @  $  2.50 ;  10  days  shelling 
corn,  @  $1.50.     Make  out  his  bill. 


270. 

Mr.  A.  L.  Jessup 


STATEMENT   OF   ACCOUNT. 

ToMBSTOXE,  Ariz.,  Dec.  16,  1907. 

In  Account  Avith  Frank  R.  Rix. 


1907 

Dr. 

Nov. 

4 

To  3  M  Brick 

7.00 

21 

00 

4 

"  6  bbl.  Cement 

2.50 

15 

00 

18 

"  4  bbl.  Lime 

1.25 

5 

00 

25 

"  4  M  Laths 

4.25 

17 

00 

25 

"  3  bbl.  Lime 

1.25 

3 

75 

25 

"  1  bbl.  Cement 
Cr. 

2 

50 

164 

25 

Nov. 

11 

By  1  ton  Hay 

14 

00 

a 

29 

"    24  lb.  Butter 
Balance  due 

.25 

6 

00 

20 

00 

$44 

25 

LEDGER  ACCOUNT. 

271.  The  foregoing  statement  of  account  is  taken  from 
the  ledger  of  Frank  B.  Bix.  Credit  transactions  are  gen- 
erally noted  in  a  day  book.  From  this  book  they  are  trans- 
ferred to  the  ledger,  in  which  all  the  accounts  of  an  indi- 
vidual are  brought  together. 

272.  The  following  shows  the  ledger  page  containing 
Mr.  Jessup's  account : 


124 


Dr. 


Arithmetic 


A.  L.  Jessup. 


Cr. 


1!)0T 

Nov. 

4 
4 

To  3  M  Brick 
"  6bbl.  Cement 

21 
15 

Nov. 

1907 
11 
29 

By  1  ton  Hay 
"241b.  Butter 

14 
6 

( ( 

18 

"  4  bbl.  Lime 

5 

(( 

25 

"  4  M  Laths 

17 

<i 

25 

"  3  bbl.  Lime 

3 

75 

(( 

25 

"  1  bbl.  Cement 
To  Balance 

2 
64 

44 

50 
25 

25 

Nov. 

30 

By  Balance 

44 
64 

25 
25 

Dec. 

1 

273.  In  the  ledger,  all  charges  against  the  person  appear 
on  the  left,  and  are  called  debits  ;  the  sums  he  pays  or  the 
goods  he  supplies  are  entered  on  the  right,  and  are  called 
credits.  The  difference  between  the  totals  of  the  two  col- 
umns shows  the  amount  due.  Since  the  debit  total:  is  the 
greater,  Mr.  Jessup  owes  Mr.  Eix  the  difference. 

The  account  is  balanced  at  regular  intervals  by  adding 
each  column  and  ascertaining  the  difference,  which  is  ^  44.25 
in  this  case.  This  is  written  in  red  ink  in  the  column  hav- 
ing the  smaller  total,  and  is  written  in  black  ink  in  the 
opposite  column.  Mr.  Jessup  on  Dec.  1, 1907,  owes  Mr.  Rix 
$  44.25,  which  is  the  first  entry  for  the  month. 


RECEIPTS. 


274. 


Albuquerque, 

N.M., 

May  25, 

1908. 

Received 

of  Lee  F.  Hanmer 

One  Hundred  Forty-three  ^% 

Dollars 

in  full  of  account  to  date. 

f  143^^1- 

M. 

E.  Williams; 

Miscellaneous  Drills 


125 


Receipts  are  all  of  the  same  general  form.  The  one  given 
above  is  "a  receipt  "  in  full,"  showing  that  Mr.  Hannier  has 
settled  all  his  indebtedness  iboMr.  Williams  to  May  25, 1908. 
If  Mr.  Hanmer  owed  Mr.  Williams  more  than  $143.75,  the 
fourth  line  of  the  latter's  receipt  would  read  "  on  account," 
which  indicates  that  there  is  an  unpaid  balance. 

A  rent  receipt  is  of  the  same  form,  except  as  to  the  fourth 
line,  which  should  read  "  in  full  of  rent  of  premises,  No.  • — 
St.,  to  May  31,  1908. 

MISCELLANEOUS   DRILLS. 


275.   Sight  Exercises. 

Multiply : 

1.   28  X  99. 

4. 

28 

x24. 

28  hundred  -  28. 

700  -  28. 

2.    28x99i. 

5. 

28 

Y  241 

28  hundred  -  14. 

700  -  14. 

3.    28  X  99f . 

6. 

28 

X24J. 

28  hundred  -  7. 

700  -  7. 

7.    28x50 

14. 

88 

X 

371 

21.   88  X  9^ 

8.   28  X  49 

15. 

88 

X 

37 

22.    24x191 

9.    28x491 

16. 

88 

X 

361 

23.    22x391 

10.    28x49f 

17. 

88 

X 

371 

24.    48x24^ 

11.    28x75 

18. 

88 

X 

871 

25.   88  X  125 

12.   28x741 

19. 

88 

X 

861 

26.   88x375 

13.   28x74f 

20. 

88 

X 

iH 

27.    88x625 

276.   "Written  Exercises. 
Note,    Do  not  write  unnecessary  figures. 
1.       486x31  2.   486 


1458 
15066  Ans, 


1458 
6318 


X  13 
Ans, 


126  Arithmetic 


♦ 


3.   486  x313  4.   1875x99 

1458  187500 

1458  185625  Ans. 


152118  Ans. 

In  obtaining  the  foregoing  results  a  line  has  been  saved  by  omitting 
to  write  the  multiplicand. 

5.  243x71  '  11.  579x213 

6.  243x17  12.  579x312 

7.  864x18  13.  385x99 

8.  864x81  14.  385x999 

9.  579x321  15.  472x999 
10.  579x123  16.  857x99 

17.  276 

X  243 


828  product  by  3  units 
828  X  8  tens  6624    product  by  24  tens 

Ans.  67068  product  by  243 

18.  276 

x324 


828       product  by  3  hundreds 
828  X  8     6624  product  by  24  units 

A71S.  89424  product  by  324 

Since  24  is  8  times  3,  we  can  obtain  the  results  in  the  foregoing 
examples  by  employing  only  two  partial  products. 

19.  415x426  23.  275x459 

20.  873x642  24.  727x273 

21.  594x742  25.  813x954 

22.  986x427  26,  586  x  45^ 


Miscellaneous   Drills  127 


27. 

157 

X3f 

471     product  by  3 

471  H-  8     58|-  product  by  | 

Ans.  529J  product  by  3f 

28. 

492 

X^ 

1968     product  by  4 

1968  H-  5    393f  product  by  i 

Ans.  2361f  product  by  4f 

29. 

265  x3f                              32.    256  X  63^ 

30. 

976  X  44                               33.   525  x  7^ 

31. 

151  X  5|                               34.    548  X  8f 

35. 

889 

X241- 

22225     product  by  25 

Deduct       444  i^  product  by  ^ 

Ans,  217801  product  by  24i 

• 

36. 

983 

X19J 

19660     product  by  20 

Deduct       122J  product  by  \ 

Ans.   195371  product  by  19J 

37. 

147  X  991                            42.    227  X  79f 

38. 

357  X  294                            43.    670  x  49i 

39. 

826x99f                            44.   659x89 J 

40. 

980x39f                            45.    768x49f 

41. 

347  X  99J                            46.    538  x  19J 

128  Arithmetic 

277.    Miscellaneous  Oral  Problems. 

1.  At  the  rate  of  2 J  miles  per  hour,  how  long  will  it 
take  a  man  to  walk  5f  miles  ? 

2.  At  2\%  an  agent  receives  %  15  for  collecting  a  bill. 
How  much  does  he  remit  to  his  employer  ? 

3.  At  what  rate  per  cent  will  any  principal  double  it- 
self in  15  years  ? 

4.  One  and  one  half  times  40  is  ^^  of  what  number? 

5.  What  per  cent  of  6f  is  4  ? 

6.  What  per  cent  of  4  is  6f  ? 

7.  Bought  41  dozen  handkerchiefs  at  25  cents  each  and 
sold  them  at  33^  cents  each.     What  did  I  gain  ? 

8.  After  spending  |  of  my  money  and  giving  away  \ 
of  the  remainder  I  had  %  60.     What  did  I  have  at  lirst  ? 

9.  A  man  paid  y^^  of  his  money  for  clothes,  $  30  for 
rent,  and  had  \  of  his  money  left.  How  much  money  had 
he  at  first  ? 

10.  A  dealer  marked  a  pair  of  shoes  25%  above  cost.  If 
he  takes  off  10%  of  the  marked  price,  what  per  cent  does  he 
gain  ? 

11.  What  is  the  cost  of  2250  pounds  of  coal  at  $  6  per 
ton  of  2000  pounds  ? 

12.  The  owner  of  a  house  pays  $  45  for  insurance  for  3 
years  on  %  3000.     What  is  the  rate  per  cent  per  year  ? 

13.  How  long  will  it  take  $  2o  to  earn  $  5  at  5%  ? 

14.  What  must  be  paid  for  the  use  of  $  400  for  1  year 
1  month  15  days  at  6%  ? 

15.  %  832  -V- 104%  equals  what  ? 

16.  When  the  cost  is  \  of  the  selling  price,  what  is  the 
gain  per  cent  ? 

17.  When  the  selling  price  is  |  of  the  cost,  what  is  the 
loss  per  cent  ? 


Miscellaneous   Drills  129 

18.  The  cost  of  |  of  anything  is  what  part  of  the  cost  of 
one  half  of  it  ? 

19.  A  sum  of  money  loaned  at  3^%,  amounts  in  6  years 
to  $  847.     How  much  was  loaned  ? 

20.  What  must  I  pay  for  4%  stock  to  get  5%  on  the 
investment  ?     (No  brokerage.) 

21.  If  I  of  the  value  of  a  field  is  $  200,  what  is  f  of  the 
value  ? 

22.  What  per  cent  of  |  is  f  ? 

23.  What  per  cent  of  |  is  |  ? 

24.  A  90-day  note  for  $  600  is  made  and  discounted  to-day 
at  6%.     Find  the  proceeds. 

25.  How  many  bolts  can  be  cut  from  an  iron  rod  7|-  feet 
long,  J  inch  being  used  for  each  bolt  ? 

26.  How  much  is  received  for  a  ton  of  coal  sold  at  30 
cents  per  basket  of  80  pounds  ? 

27.  Find  the  value  of  1122  sheep  at  $  3  each. 

28.  What  part  of  8  is  i  ? 

29.  What  is  received  for  an  article  sold  at  $  5  less  25%? 

30.  A  dealer  sold  an  article  for  $  8.10,  thereby  losing  10% . 
At  what  selling  price  would  he  have  gained  10%  ? 

31.  A  and  B  have  an  equal  number  of  cows ;  A  sells  25% 
of  his  to  B.  The  number  B  then  owns  is  what  per  cent 
greater  than  the  number  A  has  left  ? 

32.  At  4  bushel  to  a  cubic  foot,  how  many  cubic  feet  will 
contain  80  bushels  ? 

33.  If  goods  costing  $  240  are  sold  at  an  advance  of  125%, 
what  is  the  selling  price  ? 

34.  A  63-gallon  cask  is  f  full.  What  part  of  the  cask  is 
empty  after  10|  gallons  are  drawn  off  ? 

35.  Multiply  33.331  by  18.9. 


130  Arithmetic 

36.  At  |-  bushel  to  a  cubic  foot,  find  the  capacity  in  bushels 
of  a  bin  7  ft.  by  21  ft.  by  2  ft. 

37.  Find  the  area  of  a  field  125  rods  long  by  72  rods  wide. 

38.  In  what  time  will  a  sum  of  money  double  itself  at  41% 
simple  interest  ? 

39.  A  storekeeper  sold  12  dozen  eggs  for  $  1.80,  gaining 
25%.     What  did  they  cost  per  dozen  ? 

40.  How  many  square  yards  in  a  piece  of  ground  44  yards 
wide  and  110  yards  long  ? 

278.    Miscellaneous  Written  Problems. 

1.  Divide  30  and  6  hundredths  by  the  sum  of  sixteen 
hundredths  and  two  thousandths. 

2.  The  annual  expenses  of  a  school  district  are: 
Teacher's  wages  $  495 

Fuel  51 

Janitor's  wages        36 

Eepairs  72 

Books  64 

Miscellaneous  92 

What  will  be  the  school  tax  of  Mr.  Pennea,  whose  property 
is  assessed  at  $  3240,  the  assessed  value  of  the  district  being 
$40,500? 

3.  Write  in  words ;  1409.0071. 

4.  A  man  buys  a  piece  of  property  for  $85,010.  What 
monthly  rental  must  he  charge  to  net  7^  %  on  this  invest- 
ment and  his  expenses  for  repairs,  taxes,  and  insurance, 
which  amount  to  41  %  per  year  on  the  cost  of  the  property  ? 

5.  What  is  the  exact  interest  on  $  730  from  July  1, 1908 
to  Sept.  23,  1908,  at  5i  %  ? 

6.  The  amount  of  a  certain  principal  for  six  years  is 
$  650  and  the  interest  is  three  tenths  of  the  principal.  Find 
the  principal  and  the  rate  per  cent. 


Miscellaneous  Drills  131 

7.  A  and  B  set  out  from  the  same  place  and  travel 
in  opposite  directions,  A  starting  2i  hours  before  B.  A 
travels  4|  miles  per  hour  and  B  travels  34  miles  per  hour. 
How  far  apart  will  they  be  when  B  has  traveled  5  hours 
and  20  minutes  ? 

8.  If  a  man  sells  -f  of  his  farm  for  what  |  of  it  cost,  what 
per  cent  does  he  gain  or  lose  by  the  transaction  ? 

9.  Find  the  rate  at  which  f  900  in  3  years  8  months  will 
yield  $  298  interest. 

10.  Reduce  to  a  whole  or  a  mixed  number: 

631  -!■  (3  of  7^) 
(|ofl6i)-(fof  31)* 

11.  What  will  be  the  cost  of  1575  pounds  of  hay  at  $5.50 
per  ton  ? 

12.  A  man  bought  a  horse  and  a  wagon  for  $  560,  and  two 
fifths  of  the  cost  of  the  wagon  was  equal  to  two  thirds  of  the 
cost  of  the  horse.     What  did  he  pay  for  each  ? 

13.  A  stack  of  hay  will  last  a  cow  8  weeks  and  a  horse  6 
weeks.     How  many  days  would  it  last  both  ? 

14.  A  man  left  j\  of  his  property  to  his  wife,  |  of  the  re- 
mainder to  his  son,  and  the  balance,  $4000,  to  his  daughter. 
What  was  the  value  of  his  property  ? 

15.  Sold  a  horse  for  40%  advance  on  its  cost  and  with 
the  money  bought  another.  The  latter  was  sold  for  $224, 
which  was  20%  less  than  its  cost.  What  was  the  cost  of 
the  first  horse  ? 

16.  Divide  $  2380  between  A  and  B  so  that  |  of  A's  share 
will  be  equal  to  B's. 

17.  If  6  be  subtracted  from  each  term  of  the  fraction  ^-^, 
by  what  per  cent  will  the  value  of  the  fraction  be  increased 
or  diminished  ? 


132  Arithmetic 

18.  A  note  for  ^2000  at  6%  dated  Jan.  3,  1907,  has  the 
following  indorsement :  July  3, 1907,  Received  $  800.  Find 
the  balance  due  to-day. 

19.  A  drover  buys  15  horses  at  $125  each  and  25  cows  at 
$48  each.  He  loses  $210  on  the  sale  of  the  horses,  but  he 
obtains  from  the  sale  of  the  cows  a  sum  sufficient  to  yield 
him  a  profit  of  5%  on  the  whole  investment.  What  was 
the  average  price  received  for  the  cows  ? 

20.  How  much  must  be  invested  in  5%  bonds  at  115^, 
brokerage  i%,  to  secure  an  annual  income  of  $800  ? 

21.  A,  B,  and  C  can  do  together  J  of  a  piece  of  work  in 
a  day.  A  alone  can  do  y^  of  it  in  a  day,  and  B  alone  ^  of  it 
in  a  day.  What  part  of  it  can  C  do  alone  in  a  day  ?  How 
many  days  would  C  require  to  do  the  whole  work  ? 

22.  Goods  amounting  to  $487.50  were  bought  subject  to 
a  certain  discount  for  cash.  The  sum  paid  by  the  purchaser 
was  $477.75.     What  was  the  rate  of  discount  ? 

23.  After  selling  75%  of  a  stock  of  goods  at  25%  profit, 
a  merchant  sells  the  remainder  at  a  loss  of  33J%.  What  is 
the  gain  or  loss  per  cent  on  the  entire  stock  ? 

24.  Write  1908  in  Roman  notation. 

25.  Express  m  common  fractions  and  reduce  to  lowest 

terms:  .028^,  |%,  216%. 

26.  A  3-months  note  for  $960  is  made  and  discounted  to- 
day at  a  bank.     Find  the  proceeds. 

27.  What  is  the  value  of  my  house  if  my  annual  premium 
at  1%  on  I  of  its  value  is  $40? 

28.  A  farmer  has  40  acres  in  clover  which  average  10  tons 
per  acre  when  cut.  The  hay  weighs  48%  less  when  it 
is  placed  in  the  barn.  When  it  is  sold  8  months  later,  it 
weighs  12%  less  than  it  did  when  placed  in  the  barn. 
.What  was  the  weight  of  the  hay  at  the  time  of  sale? 


Miscellaneous   Drills  I33 

29.  Divide  $810  among  A,  B,  and  C,  so  that  A's  share 
may  equal  75%  of  B's,  and  B's  share  may  equal  20%  of  C's. 

30.  A  boy  bought  a  certain  number  of  apples  at  2  for  a 
cent  and  the  same  number  at  3  for  a  cent,  paying  50  cents 
for  all.     How  many  apples  did  he  buy  ? 

31.  A  boy  bought  120  peaches  at  3  for  a  cent  and  the 
same  number  at  2  for  a  cent.  By  selling  them  all  at  5  for  2 
cents,  how  much  did  he  lose  ? 

32.  The  cost  of  a  horse  equals  how  many  eighths  of  40% 
of  its  cost  ? 

33.  A  creditor  receives  $1.50  for  every  $4  due  him,  and 
thereby  loses  $  602.10.     What  was  the  sum  due  ? 

34.  How  many  tons  of  hay  at  $  8.50  per  ton  must  a  farmer 
give  in  exchange  for  13,750  feet  of  hardwood  flooring  at  $36 
per  1000  feet  ? 

35.  What  will  be  the  difference  in  annual  income  between 
$10,752  invested  in  3%  bonds  at  95 f  and  brokerage,  and 
the  same  sum  invested  in  4%  bonds  at  111  J  and  brokerage  ? 

36.  Which  will  yield  the  greater  annual  income,  3  per  cents 
bought  at  96,  including  brokerage,  or  4  per  cents  bought 
at  112,  including  brokerage  ?     What  per  cent  greater  ? 

38.  A  merchant  sells  an  overcoat  for  $22,  a  suit  for  $23, 
and  a  hat  for  $5.  He  gains  10%  on  the  overcoat,  15%  on 
the  suit,  and  25%  on  the  hat.  What  per  cent  of  the  total 
cost  of  the  articles  does  he  realize  on  the  entire  sale  ? 

39.  A  drover  sold  24  horses  for  $  150  each,  gaining  y%  of 
the  cost  on  12  of  them  and  losing  i  of  the  cost  on  the  re- 
mainder.    Find  the  total  gain  or  loss. 

40.  My  profit  on  a  sale  of  apples  amounting  to  $  455  was 
30%.  What  per  cent  would  I  have  lost  if  the  apples  had 
cost  $  300  more  ? 


134  Arithmetic 

41.  Find  the  price  of  a  6%  stock  that  will  yield  as  much 
income  as  4^%  stock  bought  at  par  (100).     No  brokerage. 

42.  If  three  eighths  of  a  quantity  costs  six  and  three 
fourths  dollars,  what  will  five  ninths  of  the  quantity  cost  ? 

43.  The  owner  of  a  house  insures  it  for  |  of  its  value  at 
the  rate  of  $75  for  three  years.  What  is  the  rate  per  year, 
the  house  being  worth  $12,000  ? 

44.  (a)  Write  in  figures :  twelve  hundred  (units)  and 
eight  thousand  three  ten-millionths. 

(b)  Express  in  words  :  20,000.002. 

45.  One  half  the  sum  of  2^  and  3J  is  how  many  times  one 
third  of  their  difference  ? 

46.  An  agent  sold  cotton  on  commission  at  5%,  and  in- 
vested the  proceeds  in  sugar  at  2%  commission,  which  was 
$  57.  What  was  the  cost  of  the  sugar  ?  What  did  he  re- 
ceive for  the  cotton  ? 

47.  (a)    Divide  .072  by  .00041 

(6)    Multiply  4  hundredths  by  7  hundred-thousandths. 

(c)  Divide  1  and    92   hundredths    by    16   ten-thou- 
sandths. 

48.  A  farm  mortgaged  for  40  %  of  its  cost  is  sold  for 
$  8000,  which  is  14^%  more  than  the  cost.  What  remains 
from  the  sale  after  the  mortgage  is  paid  and  one  half  year's 
interest  at  6%  ? 

49.  What  income  will  be  derived  from  $6500  invested  in 
7%  stock  at  130?     (Ko  brokerage.) 

50.  A  broker's  commission  for  selling  bonds  was  $  62.50. 
What  did  the  seller  receive,  the  rate  being  130,  less  broker- 
age of  1%  ? 

51.  At  what  rate  will  $  980  amount  in  270  days  to 
$  1011.85  ? 


Miscellaneous   Drills  135 

52.  What  is  a  merchant's  net  profit  on  goods  sold  to  the 
amount  of  $6000,  which  is  33^%  above  their  cost,  if  he 
pays  a  commission  of  10%  and  other  expenses  of  S800? 

53.  Divide  $  2600  among  three  persons  so  that  the  second 
shall  receive  33^%  more  than  the  third  and  the  first  shall 
receive  50%  more  than  the  second.  What  per  cent  does 
the  first  get  more  than  the  third  ? 

54.  After  spending  i  of  his  money,  then  |  of  the  remain- 
der, then  yi^  of  what  still  remained,  a  man  had  $  648.  What 
had  he  at  first  ? 

55.  The  quotient  is  123.4,  the  divisor  42,  the  remainder 
6.     Find  the  dividend. 

56.  On  a  house  costing  $  6000  there  are  the  following  ex- 
penses annually:  Insurance  for  |  of  its  value  at  i%  ;  taxes 
$  48;  repairs,  etc.,  $47.50.  What  rent  must  the  owner  re- 
ceive per  month  to  pay  expenses  and  6%  on  the  cost  of  the 
house  ? 

57.  Find  the  difference  between  the  exact  interest  and 
the  interest  by  the  common  method  on  $  120,000  at  6% 
from  March  1  to  July  25. 

58.  Simplify    «  ^  2  ^  ^2  X? 

Mixi       4of4 

59.  If  wool  costing  20^  per  pound  loses  20%  of  its 
weight  in  scouring,  at  what  price  per  pound  must  the  scoured 
wool  be  sold  to  gain  20  %  ? 

60.  Write  an  interest-bearing  promissory  note  for  $  500, 
dated  July  10,  1907,  maturing  to-day  with  interest  at  6%. 
Find  the  amount  due  to-day. 

61.  What  amount  invested  in  U.S.  3's  at  115,  brokerage 
i%,  will  yield  an  annual  income  of  $  1200  ? 

62.  A  dealer  imports  1250  pounds  of  cheese  at  13  f^^  per 
lb.  (duty  8^  per  lb.),  1000  gross  of  matches  at  40  ^  per  gross 
(duty  at  8^  per  gross),  and  300  pounds  of  preserved  fruit  at 


136  Arithmetic 

80^  per  lb.  (duty  1  <f  per  lb.  and  35  %).     Find  the  total  cost, 
including  duty. 

63.  If  %  324  is  paid  for  a  piano  at  40,  2h^  and  10  %  dis- 
count, what  is  the  list  price  ? 

64.  A  dealer  sold  a  piano  marked  %  520  for  $  460.  If 
the  marked  price  is  30%  above  the  cost,  what  per  cent  did 
he  gain  ? 

65.  A  and  B  together  have  $  500 ;  A  and  C  have  together 
%  600.  How  much  has  C  more  than  B  ?  If  B  and  C  have 
together  $  700,  how  much  has  each  ?     How  much  has  A  ? 

66.  A  farmer  sold  a  horse,  a  cow,  and  a  sheep.  The  price 
of  the  horse  and  the  cow  was  %  200 ;  of  the  horse  and  the 
sheep  %  160 ;  of  the  cow  and  the  sheep  %  60.  What  was  the 
price  of  each  ? 

67.  A  man  left  his  property  worth  $  20,000  to  his  wife, 
his  daughter,  and  his  son,  the  daughter's  share  being  20% 
more  than  the  son's,  and  his  wife's  share  being  50%  more 
than  the  daughter's.     What  was  the  share  of  each  ? 

68.  A  man  has  three  casks.  The  second  has  one  third 
of  the  capacity  of  the  first,  and  the  third  has  four  ninths  of 
the  capacity  of  the  first.  If  the  contents  of  the  other  two 
are  poured  into  the  first,  it  will  still  hold  8  gallons.  What 
is  the  capacity  of  each  if  the  capacity  of  the  first  is  8  gal- 
lons greater  than  the  combined  capacity  of  the  other  two  ? 

69.  A  New  York  banker  shipped  $48,665  in  gold  to  Lon- 
don to  settle  an  account  amounting  to  £  10,000.  He  paid 
\%  freight  and  1%  for  insurance.  There  was  a  loss  of  y6-% 
by  abrasion  on  $  20,000  in  $  20  gold  pieces,  of  i  %  on 
$20,000  in  $10  gold  pieces,  and  of  i%  on  the  $5  gold 
pieces,  which  constituted  the  remainder  of  the  shipment. 
What  was  the  total  cost  to  the  banker,  including  the  sum 
paid  to  replace  the  loss  by  abrasion  ? 


Miscellaneous   Drills 


137 


70.  Make  out  a  bill  to  J.  F.  Fagan  for  five  articles  bought 
from  C.  E.  Wise  &  Co.     Receipt  it. 

71.  A  real  estate  agent  sold  a  building  for  $  10,000. 
Find  his  commission  at  5%  on  the  first  $1000;  2^%  on 
$4000,  and  1\%  on  the  remainder. 

72.  Water  in  freezing  expands  10%  in  bulk.  If  a  cubic 
foot  of  water  weighs  1000  ounces,  what  will  be  the  weight 
of  a  cubic  foot  of  ice  ? 

73.  A  real  estate  agent  whose  rates  are  5%  on  the  first 
$1000,  21%  on  all  above  $1000  and  not  exceeding  $5000, 
and  li%  on  all  above  $5000,  received  a  commission  of 
$24:3r.7o  for  selling  a  farm.  What  price  was  paid  for  the 
farm  ? 

74.  Write  one  thousand  forty-five  and  nine  millionths. 
Write  seven  and  nine  thousandths. 

75.  Simplify  63i  +  (fof7i)      ^ 

^     ^  (t  of  161) -d  of  30 

76.  A  furniture  dealer  sold  two  desks  for  $  30  and  $  28 
respectively.  On  the  first  he  lost  16|%,  and  on  the  other 
he  gained  16|  %.  What  per  cent  of  loss  or  of  gain  was  there 
on  the  entire  transaction  ? 

77.  ]Mr.  Mard  bought  a  house  for  $  4500,  paying  $  900  in 
cash  and  giving  three  equal  notes  for  the  balance  payable 
in  4,  8,  and  12  months  respectively,  with  interest  at  6%. 
What  was  the  total  sum  paid,  including  interest  ? 

78.  An  agent  received  a  commission  of  $223  on  sales 
amounting  for  a  month  to  $1784.  AVhat  should  his  commis- 
sion be  the  following  month  if  he  sold  goods  to  the  amount 
of  $  1496  ? 

79.  A  dealer  bought  6  dozen  oranges  at  15  cents  per  dozen. 
He  sold  1  of  them  at  2  for  5  cents,  ^  of  them  at  1  cent  each, 
and  the  remainder  at  3  for  5  cents.  What  per  cent  did  he 
gain  ? 


138  Arithmetic 

80.  Compile  a  cdtopound-interest  table  showing  the  com- 
pound interest  on  $  1  for  1  year,  2  years,  3  years,  4  years, 
at  1%,  2%,  3%,  4%,  and  5%,  to  four  places  of  decimals. 

81.  A  miller  purchased  1200  bushels  of  wheat  through 
an  agent,  at  95^  per  bushel.  The*  total  cost  to  the  miller,  in- 
cluding freight  charges  of  $  60  and  the  agent's  commission, 
was  $  1248.     How  much  per  bushel  was  the  commission  ? 

82.  Find  the  difference  between  the  accurate  interest  and 
the  regular  interest  on  ^800  from  April  1  to  Nov.  6  at  5  %. 

83.  The  assessed  valuation  of  a  town  is  $  875,000.  The 
whole  amount  of  taxes  to  be  raised  is  $13,125.  What  is 
A's  tax,  if  his  property  is  assessed  at  $  1680  ? 

84.  Write  as  common  fractions  : 

(a)  .025.      (6)  f  of  one  per  cent.      (c)  i%.      (d)  6^%. 
Change  the  following  to  expressions  having  the  per  cent 
sign  : 

(e)  .073.      (/)  f.      (g)  .00|.      (h)  |. 

85.  A  shipper  sent  a  commission  merchant  2100  bushels 
of  wheat  to  sell,  and  instructed  him  to  invest  the  net  pro- 
ceeds in  salt  at  $2  per  barrel.  The  commission  merchant 
deducted  2%  commission  for  selling  the  wheat  and  sent 
the  shipper  980  barrels  of  salt,  for  the  purchase  of  which 
he  charged  a  commission  of  5%.  What  price  per  bushel 
was  obtained  for  the  wheat  ? 

86.  By  selling  at  20%  less  than  his  asking  price,  a  mer- 
chant makes  a  profit  of  20%.  What  per  cent  profit  would 
he  have  made  if  he  had  obtained  his  asking  price  ? 

87.  A  man  bought  a  watch  and  a  chain  for  $70.  One 
half  of  the  cost  of  the  watch  equals  two  thirds  of  the  cost 
of  the  chain.     Find  the  cost  of  each. 

88.  What  per  cent  is  gained  by  buying  at  the  rate  of  3 
yards  for  $  2  and  selling  at  the  rate  of  2  yards  for  $  3  ? 


Miscellaneous   Drills  139 

89.  A  buys  stock  at  78  and  sells  it  at  84.  B  buys  bonds 
at  70  and  sells  at  75.  A  makes  $  1000  more  than  B.  How 
much  did  A  invest  ?     (No  brokerage.) 

90.  A  bin  6  feet  long  4  fee't  wide  holds  72  bushels.  How 
deep  is  the  bin,  assuming  that  1  cubic  foot  equals  4  bushel  ? 

91.  A  dealer  sold  f  of  a  quantity  of  cloth  at  a  profit  of 
20%  and  the  remainder  at  cost.  What  per  cent  did  he 
gain  ? 

92.  A  note  for  $7200  dated  Sept.  1,  1906,  with  interest 
at  6%  bears  the  following  indorsements  : 

Jan.  2,  1907,  $1250;  July  25,  1907,  $45;  Dec.  26, 
1907,  $975;  May  1,  1908,  $600.  Find  the  amount  due 
July  3,  1908. 

93.  Reduce  .1944|-  to  a  simple  fraction  in  its  lowest 
terms. 

94.  If  18  pipes  each  delivering  6  gallons  per  minute  fill 
a  cistern  in  2  hours  and  16  minutes,  how  many  pipes,  each 
delivering  20  gallons  per  minute,  will  fill  a  cistern  7|- 
times  as  large  in  3  hours  24  minutes  ? 

95.  A  man  sells  45  shares  of  stock  at  118|-,  and  invests 
in  3^%  bonds  at  89|-,  brokerage  i%  in  each  case.  If  the 
stock  paid  5%,  by  how  much  is  his  yearly  income  changed? 
How  does  the  certainty  of  his  income  from  the  bonds  com- 
pare with  that  from  his  investment  in  stock  ? 

96.  Three  fourths  of  A's  farm  is  equal  in  value  to  two 
thirds  of  the  value  of  B's.  Together,  the  farms  are  worth 
$  13,600.     What  is  the  value  of  each  ? 

97.  A  invests  $1200  and  makes  $500  in  four  months. 
B  invests  $750  and  makes  $250  in  five  months.  Which 
makes  the  greater  per  cent  a  month  on  his  money,  and  how 
much  ? 

98.  Find  the  interest  on  $  2500  from  Sept.  13,  1905,  to 
May  4,  1908,  at  5  per  cent. 


140  Arithmetic 

99.  By  selling  an  article  marked  $60  at  a  discount  of 
20%,  a  profit  of  20%  is  made.  What  did  the  article  cost 
the  seller  ? 

100.  What  is  the  cost  of  19,250  pounds  of  coal  at  f  6  per 
ton  of  2000  pounds  ? 

101.  What  per  cent  must  be  added  to  the  cost  to  leave  a 
profit  of  20%  after  a  discount  of  25%  ? 

(3.75  +  19.867)  x  (82.5-28.425) 
10.815 
103.    Find  the  cost  of  a  cable  transfer  of  £  426  16s.  at 
14.8735  per  £. 


CHAPTER   V. 

DENOMINATE  NUMBERS;   MEASUREMENTS. 

REVIEW   OF   DENOMINATE   NUMBERS. 

279.  Preliminary  Exercises. 

English  Money. 

4  farthings  =  1  penny  (d.) 
12  pence  =  1  shilling  (s.) 
20  shillings  =  1  pound     (£') 

1.  How  many  pence  in  4  shillings  ? 

2.  Change  4  shillings  4  pence  to  pence. 

3.  Eeduce  £  21  to  shillings. 

4.  How  many  shillings  in  £  2  10s.  ? 

5.  Change  50  shillings  to  pence. 

6.  How  many  pence  in  50s.  6cl.  ? 

7.  How  many  shillings  in  £i? 

8.  Eeduce  £i  to  shillings  and  pence. 

9.  How  many  shillings  in  .25  of  £  ? 

10.  How  many  shillings  and  pence  in  .125  of  £  ? 

11.  Change  £  |  to  shillings  and  pence. 

REDUCTION  DESCENDING. 

280.  Written  Exercises. 

1.    Eeduce  £  24  17s.  6cl.  to  pence. 


£  24  =  20s. 

x24 

=  480s. 
+  17s. 

£24  17s. 

12cl   X 

497 

=    497s. 

£24  17s.  = 

=  5964(?. 
+  6fZ. 

£24  17s.  6d 

r 

=  5970d. 

-■^%  - 

141 

Ans. 


20s. 

V2d. 

£24  17s. 

M. 

497s. 

6d. 

142  •    Arithmetic 

Since  there  are  20  shillings  in  a  pound,  in  £24  there  are  24  times 
20  shillings,  or  480  shillings.  In  £24  17s.  there  are  480  shillings  +  17 
shillings,  or  497  shillings. 

Since  there  are  12  pence  in  a  shilling,  in  497  shillings  there  are  497 
times  12  pence,  or  5964  pence.  In  497s.  6d.  (£24  17s.  6d.)  there  are 
5964  pence  +  6  pence,  or  6970  pence.     A71S.  5970d. 

The  accompanying  arrangement  of  the  work  is  sug- 
gested.    Above  17s.  write  the  number  of  shillings 
(20s.)  in  £1  ;  above  6d.  write  the  number  of  pence 
Ans.   5970c?.       {12d.)  in  a  shilling. 

Multiply  20s.  by  24,  and  add  in  17s.  at  the  same 
time,  which  gives  497s.  for  the  first  step.  Multiply  I2d.  by  497  and 
add  in  6d.  for  the  final  result. 

In  practice,  however,  20  and  12  are  employed  as  the  multipliers. 

2.  Reduce  47  gal.  1  pt.  to  pints. 

4  qt.  2  pt.  Write  0  qt.  in  the  proper  place.     Above  the 

47  gal.      0  qt.  1  pt.        column  of  quarts  write  4  qt.,  the  number  of 
188  qt.  1  pt.        quarts  in  a  gallon,  and  above  the  column  of 
Ans.  377  pt.       pints  write  2  pt.,  the  number  of  pints  in  a 
quart. 
Eeduce : 

3.  £  47  15s.  lOd  to  pence. 

4.  £  33  Sd.  to  pence. 

5.  £59  16s.  to  pence. 

6.  27  gal.  1  pt.  to  pints. 

7.  14  gal.  3  qt.  to  pints. 

8.  13  bu.  2  pk.  3  qt.  to  quarts. 

9.  27  yd.  1  ft.  9  in.  to  inches. 

10.   3  da.  15  hr.  40  min.  to  minutes. 

281.    Troy  Weight. 

24  grains  (gr.)        =  1  pennyweight  (pwt.) 
20  pennyweights    =  1  ounce  (oz.) 
12  ounces  =  1  pound  (lb.) 

Troy  weight  is  used  in  weighing  gold  and  silver. 

1  lb.  troy  =  5760  grains. 

1  lb.  avoirdupois  =  7000  grains. 


Denominate  Numbers  143 

Keduce : 

1.  15  pwt.  17  gr.  to  grains. 

2.  12  ounces  troy  to  grains. 

3.  f  lb.  avoirdupois  to  grains. 

4.  1  oz.  avoirdupois  to  grains. 

5.  3  lb.  7  oz.  12  pwt.  16  gr.  to  grains. 

6.  1  lb.  troy  to  grains. 

7.  1  lb.  troy  to  fraction  of  avoirdupois  pound. 

8.  1  oz.  avoirdupois  to  fraction  of  troy  ounce. 

9.  1  lb.  avoirdupois  to  lb.,  oz.,  etc.,  troy. 
10.   2  lb.  4  pwt.  to  grains. 

REDUCTION  ASCENDING. 
282.   Preliminary  Exercises. 
Reduce : 

1.  48  pence  to  shillings. 

2.  52  pence  to  shillings  and  pence. 

3.  120  shillings  to  £. 

4.  135  shillings  to  £,  etc. 

5.  240  pence  to  £. 

6.  250  pence  to  £,  etc. 

7.  2s.  6d.  to  the  fraction  of  a  £. 

8.  7s.  6d.  to  the  decimal  of  a  £. 

9.  480  grains  to  ounces. 

10.  490  grains  to  ounces  and  grains. 

11.  60  grains  to  the  decimal  of  an  ounce. 

12.  .125  lb.  to  pennyweights  and  grains. 

13.  I  lb.  to  pennyweights  and  grains. 
14.-  96  quarts  to  bushels. 

15.    125  pounds  to  the  fraction  of  a  ton. 


144  Arithmetic 

283.    Written  Exercises. 

1.  Eeduce  5970  cZ.  to  higher  denominations. 

Above  5970fZ.  write  12d.,  the  equivalent  of 
Is.  in  pence.     Dividing  5970d.  by  12cZ.,  we  ob- 
tain a  quotient   of   497,  the  number  of  shil- 
lings, and  a  remainder  of  Qd. 
£  24        17s.  Qd.         ^j^g  number  of  pounds  sterling  in  497s.  is 

obtained  by  dividing  by  20s.,  which  gives  24  and  a  remainder  of  17s. 

2.  Reduce  377  pints  to  higher  denominations. 

4  qt.       2  pt. 
377  pt. 


20s. 

12d. 

5970d. 

497s. 

6d. 

188  qt.       1  pt. 


47  gal.  1  pt.     Ans. 

Eeduce  to  higher  denominations  : 

472  qt.  (dry). 
288  gr.  to  fraction  of  lb. 
576  gr.  to  decimal  of  lb. 
200d  to  fraction  of  £. 
15.   17s.  6d.  to  decimal  of  £. 

1  hr.  30  min.  to  fraction  of  da. 
13  hr.  30  min.  to  decimal  of  da. 
10.   356  pt.  (liquid).       18.    3  qt.  to  decimal  of  bu. 

OPERATIONS  IN  DENOMINATE  NUMBERS. 
284.   Sight  Exercises. 


3. 

5400  grains. 

11. 

4. 

3600  pwt. 

12. 

5. 

4000  grains. 

13. 

6. 

2740  pence. 

14. 

7. 

1500  pence. 

15. 

8. 

3750  minutes. 

16. 

9. 

3750  seconds. 

17. 

1.        24  lb.  8  oz.          2.      49  lb. 

-f-  24  lb.  8  oz.              -  24  lb.  8  oz. 

3.    24  lb.  8  oz. 

X  2 

4.    2)49  lb.               5.    24  lb.  8  oz.)  49  lb. 

7.        3  qt.  1  pt.           8.      8  gal. 

-f  5  qt.  1  pt.               —  3  gal.  3  qt. 

6.   i  of  17  1b. 

9.    10  gal.  3  qt. 
x3 

10.   4)£20  16s.  8d                    11.    iof9 

oz.  15  pwt.  18  gr. 

Denominate  Numbers  145 

285.   Written  Exercises. 

1.    Add  £  24  16s.  Sd.,  £  37  9c?.,  17s.  6icL,  £  42  16s. 

24     16s.  8  cl  Sd.  +  9d.  +  6^d.  =  2S^d.   =  Is.  llifZ.     Write 

37  9      •  ll^d.  and  carry  Is. 

17  6i  Is.-f  16s.+17s.  +  16s.  =  50s.  =£2  10s.    Write 

42     16  10s.  and  carry  £  2. 


£105     10s.       Hid         £2 +£42 +£37 +  £24  =  £105. 

Add : 

2.  £  2  10s.  6cl,  17s.  Scl,  £  24  13s. 

3.  14  lb.  6  oz.,  27  lb.  10  oz.,  17  oz.,  19  lb.  12  oz.  (avoirdu- 
pois). 

4.  3  bu.  2  pk.  7  qt.,  5  bii.  3  pk.,  2  pk.  6  qt. 

5.  13  gal.  3  qt.  1  pt.,  17  gal.  2  qt.,  3  qt.  1  pt.,  25  gal. 

6.  14  lb.  6  oz.  17  pwt.,  29  lb.  8  oz.,  10  oz.  8  pwt.,  4  lb. 
15  pwt. 

7.  7  yd.  1  ft.  9  in.,  18  yd.  2  ft.  10  in.,  25  yd.  1  ft.  6  in. 

8.  £  33  17s.  6d.,  £  24  15s.  Sd.,  £  19  15s.  lie?. 

9.  3  oz.  13  pwt.,  8  oz.  15  pwt.  18  gr.,  19  pwt.  10  gr.,  15  gr. 

10.  24  bu.  3  pk.,  2  bu.  2  pk.  2  qt.,  3  pk.  7  qt.,  4  qt. 

11.  From  32  bu.  1  pk.  1  qt.  take  14  bu.  3  pk.  5  qt. 

32  bu.    1  pk.    1  qt.  Since  5  qt,  is  greater  than  1  qt.,  we 

14  bu.    3  pk.    5  qt.  borrow  1  pk.,   or  8  qt.,  making  9  qt. 

17  bu.    1  pk.    4  qt.  5  qt.  from  9  qt.  leaves  4  qt.,  etc. 

12.  £  22  10s.  6d.  -  £  4  17s.  8c?. 

13.  23  bu.  1  pk.  1  qt.  -  15  bu.  3  pk.  6  qt. 

14.  145  gal.  1  qt.  -  24  gal.  3  qt.  1  pt. 

15.  23  lb.  9  oz.  5  pwt.  -  5  lb.  10  oz.  18  pwt. 

16.  115  oz.  10  pwt.  12  gr.  —  24  oz.  15  pwt.  20  gr. 

17.  240  yd.  1  ft.  6  in.  -  115  yd.  2  ft.  9  in. 

18.  £  100  Cxi  -  £  20  15.S-.  9d. 

19.  G3  gal.  -  42  gal.  2  qt.  1  pt. 


146  Arithmetic 

20.  123  yd.  7  in.  -  60  yd.  2  ft.  9  in. 

21.  Multiply  4  wk.  18  hr.  30  min.  by  5. 

4  da.    18  hr.    30  rain.  30  min.  x  5  =  150  min.  =  2  hr.  30  min. 

5 Write  30  min.  and  carry  2  hr. 

23  da.   20  hr.    30  min.  18  hr.  x  5  =  90  hr.     Add  2  hr.,  mak- 

ing 92  hr.,  which  equals  3  da.  20   hr. 

Write  20  hr.  and  carry  3  da. 

4  da.  X  5  =  20  da.    Add  3  da.,  making  23  da. 

22.  3  wk.  4  da.  12  hr.  x  6. 

23.  14  bu.  2  pk.  7  qt.  x  8. 

24.  £15  17s.  6d.  X  7. 

25.  3  oz.  15  pwt.  12  gr.  x  9. 

26.  17  gal.  1  qt.  1  pt.  X  4. 
.     27.  153  yd.  2  ft.  9  in.  x  3. 

28.  £43  9s.  M.  x  12. 

29.  7  oz.  12  pwt.  10  gr.  x  5. 

30.  24  bu.  2  pk.  3  qt.  x  11. 

31.  Divide  23  da.  20  hr.  30  min.  by  5. 

5)23  da.    20  hr.   30  min.  23  da.  h-  5  =  4   da.   and  3   da.  re- 

4  da.    18  hr.   30  min.       mainder.     Write  4  da. 

3  da.  =  72  hr.     72  hr.  +  20  hr.  =  92 
hr.     92  hr.  -f-  5  =  18  hr.  and  2  hr.  remainder.     Write  18  hr. 

2  hr.  =  120  min.     120  min.  +  30  min.  =  150  rain.     150  min.  -f-  5  = 
30  min.     Write  30  rain. 

32.  247  days  -r-  8. 

33.  139  troy  ounces  ^  6. 

34.  i  of  23  da.  20  hr.  30  min. 

35.  I  of  11  da.  22  hr.  15  min. 

36.  I  of  175  bu.  3  pk.  4  qt. 

37.  f  of  £27  17s.  4d. 
38o  .75  of  £55  14s.  8d 
39.  .125  of  43  bushels. 


Denominate  Numbers  147 

40.  .375  of  £250. 

41.  Divide  23  da.  20  hr.  30  min.  by  4  da.  18  hr.  30  min. 

When  the  divisor  and  the  dividend  are  concrete  numbers,  they  must 
have  the  same  denomination.  Reducing  both  terms  to  minutes,  we 
have,  23  da.  20  hr.  30  min.  =  34350  min. 

4  da.  18  hr.  30  min.  =  6870  min. 
34350  min.  --  6870  min.  =  5. 
That  is,  34350  minutes  contains  6870  minutes  5  times. 

42.  Divide  £106  by  £6  12s.  6d 

£106  =  25440d. 
£6  12s.  Qd.  =  1590(Z. 
25440cZ  -^  1590d  =  16.     Ans. 

The  divisor  may  be  reduced  to  £  6f ,  in  which  case  the  problem 
becomes, 

£  106  --  £6f  =  106  -^  6f  =  106  --  ^/  =  106  x  3*3.     Cancel.       . 

Note.  When  the  divisor  and  the  dividend  are  concrete,  the 
quotient  is  abstract. 

43.  £30 --£3  15s. 

44.  £  30  --  2s.  M, 

45.  1  da.  -r- 1  hr.  36  rain. 

46.  2  oz.  ^  1  pwt.  16  gr. 

47.  23  gal.  1  qt.  1  pt.  -=- 1  gal.  1  qt.  1  pt. 

48.  12  yd.  2  ft.  6  in.  -- 1  yd.  2  ft.  6  in. 

49.  46  bu.  4  qt.  -f-  3  bii.  3  pk.  3  qt. 

50.  22  hr.  32  min.  30  sec.  -^  2  hr.  15  min.  15  sec. 

286.    Oral  Problems. 

1.  What  is  the  cost  in   English   money  of  8  yards  of 
silk  at  2s.  6c?.  per  yard  ? 

2.  Eight    ounces    avoirdupois    is    what   fraction   of  4 
pounds  ? 

3.  How  many  flag-stones  8  feet  long  will  be  required  for 
a  walk  120  yards  long  ? 


148  Arithmetic 

4.  Find  the  time  in  hours  and  minutes  from  8.30  a.m. 

to  1.15  P.M. 

5.  A  ten-dollar  gold  piece  weighs  lOJ  pennyweights. 
How  many  grains  does  it  weigh  ? 

6.  How  many  strips  of  carpet  J  yard  wide  will  be  re- 
quired to  cover  a  floor  18  feet  wide  ? 

7.  Mr.  Tully  sells  50  quarts  of  milk  a  day.  How 
many  gallons  does  he  sell  in  30  days  ? 

8.  At  60  lb.  per  bushel,  how  many  bushels  and  pecks  will 
there  be  in  375  pounds  of  wheat  ? 

9.  A  road  is  4  rods  wide.     What  is  its  width  in  feet? 
10.    How  many  inches  in  25  yards  ? 

287.    "Written  Problems. 

1.  How  much  is  the  profit  on  126  bushels  of  potatoes 
bought  at  80  cents  a  bushel  and  sold  at  15  cents  a  half  peck  ? 

2.  Find  the  average  height  of  20  boys,  2  of  whom 
measure  5  ft.  5  in.  each  ;  4,  5  ft.  6  in.  each ;  6,  5  ft.  7  in. 
each;  and  8,  5  ft.  8  in.  each. 

3.  A  horse  is  fed  6  quarts  of  oats  per  day.  How  many 
bushels,  pecks,  and  quarts  will  he  eat  in  July? 

4.  A  field  is  80  rods  long  and  20  rods  wide.  How 
many  feet  of  wire  will  be  required  to  inclose  it  with  a  fence 
four  strands  high  ? 

5.  A  seed  dealer  made  up  500  half-pint  packages  of 
beans.     How  many  bushels,  pecks,  etc.,  were  required? 

6.  How  many  ^-pt.  bottles  will  be  needed  to  hold  63 
gallons  of  catsup? 

7.  At  $16  per  ton,  find  the  cost  of  14  bales  of  hay  aver- 
aging 175  pounds  each. 

8.  At  $40  per  month,  what  is  the  total  rent  of  a  house 
for  5  years  7  months  15  days  ? 


Standard  Time 


149 


9.   What  do  I  pay  for  the  use  of  a  certain  sum  of  money 
for  3  years  8  months  6  days  at  the  rate  of  $  60  per  year  ? 

10.  A  dealer  buys  500  tons  of  coal  at  $5  per  ton  of  2240 
pounds.  He  sells  it  at  $  6  per  ton  of  2000  pounds.  What 
is  his  profit  after  deducting  $375  for  freight,  cartage,  etc.? 

STANDARD   TIME. 

288.  Problems  in  longitude  and  time  have  but  little  prac- 
tical value  except  for  navigators,  owing  to  the  almost  uni- 
versal employment  of  standard  time. 

289.  Although  no  two  places  not  on  the  same  meridian 
can  have  the  same  solar  time,  tlie  railroads  of  the  United 
States  and  Canada  agreed  in  1883  that  time  belts  should  be 
established  approximating  15°  in  width,  and  that  within 
these  belts  the  time  should  be  that  of  60°,  75°,  90°,  105°, 
and  120°  w^est  longitude,  respectively ;  that  is,  4,  5,  6,  7, 
and  8  hours,  respectively,  earlier  than  the  time  of  Green- 
wich. These  time  belts,  beginning  on  the  east,  are  known 
as  the  Atlantic,  Eastern,  Central,  Mountain,  and  Pacific. 
The  boundaries  of  the  belts  are,  as  nearly  as  may  be,  7^° 
on  each  side  of  the  meridian,  though  each  railroad  changes 
its  time  at  some  important  point  on  its  own  line. 
Many  of  the  roads  passing  through  Buffalo  change  from 
eastern  to  central  time  at  that  point ;  the  Grand  Trunk,  how- 
ever, makes  its  change  at  Sarnia,  Canada. 

290.  Greenwich  time  is  used  throughout  England,  Hol- 
land, Belgium,  and  Spain. 

Germany  employs  Central  European  time,  which  is  one 
hour  later  than  that  of  Greenwich.  This  is  also  the  time  of 
Austria,  Hungary,  Italy,  Switzerland,  Xorway,  and  Sweden. 

Prance,  Algiers,  and  Tunis  hold  to  the  time  of  Paris,  9 
minutes  faster  than  that  of  Greenwich. 

European    Russia,    Turkey    in    Europe,    Roumania,    and 


I50 


Arithmetic 


Bulgaria  use  the  time  of  St.. Petersburg,  which  is  2  hours 
1  minute  13  seconds  faster  than  Greenwich  time. 
'    Japan  time  is  9  hours  later  than  Greenwich.     Australia 
has  several  time  belts. 

Note.   It  is  suggested  that  but  little  attention  be  given  to  problems 
involving  solar  time. 

LONGITUDE   AND   TIME. 

291.  Circular  Measure. 

60  seconds  (")    =  1  minute  (') 
60  minutes  =  1  degree  (°) 

360  degrees         =  1  circle, 

292.  By  the  longitude  of  a  place  is  meant  the  distance  in 
degrees  measured  on  the  equator  between  the  meridian  of 

the  place  and  the  prime  merid- 
ian. 

The  meridian  of  a  place  is 
the  half  of  the  great  circle 
passing  through  the  place  and 
terminating  at  the  poles. 
The  prime  meridian  is  gen- 
erally taken  as  the  meridian 
passing  through  Greenwich, 
England.     Its  longitude  is  0°. 

The  longitude  of  A  is  75°  east ; 
that  of  B,  60°  west ;  of  M,  45°  west ;  of  N,  30°  east. 

293.  Oral  Exercises. 

Find  the  difference  in  longitude  between : 

1.  A  and  G  5.   A  and  B  9.   B  and  N 

2.  jB  and  (^  6.    ^  and  Jf  10.   M  and  JSf 

3.  M  and  G  1.   A  and  N 

4.  ^and  G  8.   ^  and  M 

294.  Since  the  earth  revolves  from  west  to  east,  making  a 
complete  revolution,  o60°,  in  24  hours,  the  sun  appears  to 


Longitude  and  Time 


151 


travel  from  east  to  west  at  the  rate  of  15°  in  1  hour.  A 
being  75°  east  of  G,  it  will  be  noon  at  ^  5  hours  earlier 
than  at  G. 

295.    Find    the    difference    in    sun  time  between  places 
whose  longitudes  are,  respectively  : 


6.  45°  W.  and  0° 

7.  60°  W.  and  30°  E. 

8.  75°  E.  and  60°  \V, 

9.  60°  W.  and  45°  W. 
10.  30°  E.  and  0° 


1.  75°  E.  andO° 

2.  45°  W.  and  30°  E. 

3.  75°  E.  and  30°  E. 

4.  60°  W.  and  0° 

5.  75°  E.  and  45°  E. 
Note.     The  more  easterly  place  has  the  later  time. 

296.    Oral  Exercises. 

When  it  is  noon  at  G,  find  the  time  at : 

1.  A,  75°  E.  3.    J/,  45°W. 

2.  B,  60°  W.  4.    ^Y.  30°  E. 

"When  it  is  noon  at  A,  75°  E.,  find  the  time  at : 

5.  G,0°  7.    3f,  45°W. 

6.  B,  60°  W.  8.    N,  30°  E. 
When  it  is  noon  at  B,  60°  W.,  find  ,the  time  at : 


9.    6^,0° 

11. 

M,  45°  W. 

10.    A,  75°  E. 

12. 

jsr,  30°  E. 

^ind  the  longitude 

or  the  time : 

Longitude  of  X. 

Lt 

)NGITUI)E   OF 

Y. 

Time  at  X. 

Time  at   Y. 

13.         ? 

45°  E. 

9  A.M. 

1  P.M. 

14.   75°  W. 

9 

2  P.M. 

12  m. 

15.   30°E. 

45°  W. 

? 

3  p.m. 

16.    60°  E. 

45°  E. 

7  A.M. 

? 

17.   75°  E. 

9 

4  P.M. 

NOON- 

18.       ? 

60°  W. 

5  A.M. 

1  A.M. 

152  Arithmetic 


Longitude  of  X.     Longitttde  of  Y.      Time  at  X.  Time  at  Y. 

19.  30°  E.  45°E.  11  A.M.  ? 

20.  75°  E.  30°  W.  ?  7  p.m. 

SOLAR  TIME. 

297.  To  determine  the  longitude  of  a  vessel,  the  captain 
notes  the  time  shown  by  his  chronometer,  when  his  observa- 
tion of  the  sun  shows  the  ship's  time  to  be  exactly  noon. 
The  chronometer  gives  the  time  at  Greenwich.  If  the 
chronometer  marks  exactly  1  o'clock,  the  vessel  is  15°  west 
of  the  meridian  of  Greenwich ;  if  it  marks  10  o'clock,  the 
vessel  is  30°  east  of  the  meridian  of  Greenwich. 

298.  "Written  Exercises. 

1.  Find  the  longitude  of  a  vessel  when  the  chronometer 
indicates  12  minutes  30  seconds  past  2  at  the  moment  the  cap- 
tain's observation  of  the  sun  determines  that  it  is  exactly 
noon. 

A  difference  of  1  hour  in  time  makes  a  difference  of  15°  of  longi- 
tude; a  difference  of  1  minute  of  time  makes  a  difference  of  15'  of 
.longitude;  a  difference  of  1  second  of  time  makes  a  difference  of  15" 
of  longitude, 

15°  X  2  (the  number  of  hours)  =  30° 
15'  X  12  (the  number  of  minutes)  =  3° 
15"  X  30  (the  number  of  seconds)  =        7'  30" 


Difference  in  longitude  =  33°  7'  30" 

In  practice,  the  difference  in  time  is  2°  12'  30" 

written  as  degrees,  minutes,  and  seconds  x  15 


and  multiplied  by  15.  Ans.  33°  7'  30"  west. 

As  the  vessel's  time  is  earlier  than  that  of  Greenwich,  the  vessel  is 
west  of  Greenwich. 

To  find  the  difference  in  longitude,  write  the  time  difference 
as  degrees,  minutes,  and  seconds,  and  midtiiily  hy  15. 

2.    Find  the  difference  in  longitude  between  two  places 
whose  difference  in  time  is  4  hr.  15  min-  25  sec. 


Longitude  and  Time  153 

3.  The  difference  in  time  between  two  places  is  3  hr. 
20  min.     Find  the  difference  in  longitude. 

4.  When  it  is  12  noon  at  Greenwich,  it  is  7:30  a.m.  at 
B.     What  is  the  longitude  of  ^  ? 

5.  It  is  noon  at  X  when  it  is  7:30  a.m.  at  Greenwich. 
Find  the  longitude  of  X. 

6.  Find  the  difference  in  time  between  two  places  which 
differ  in  longitude  33°  7'  30". 

Since  the  difference  in  longi- 

15)  3.3  hr.       7  min.     30  sec.         tude  is  obtained  by  multiply- 

Ans.  2  hr.     12  min.     30  sec.        ing  the  time  difference  by  15, 

the  latter  is  obtained  by  divid- 
ing the  longitude  difference  by  15,  the  number  of  degrees  of  longi- 
tude being  changed  to  hours,  etc. 

To  find  the  difference  in  time,  ivrite  the  loyigitude  difference 
as  hours,  minutes,  and  seconds,  and  divide  by  15. 

7.  The  difference  in  longitude  between  two  places  is  48° 
40'  15".     Find  the  difference  in  time. 

8.  When  it  is  noon  at  Greenwich,  what  is  the  time  at 
a  place  in  longitude  74°  36'  30"  east? 

9.  Find  the  time  of  a  place  in  longitude  48°  27'  45" 
west  when  it  is  12  m.  at  Greenwich. 

10.  When  it  is  9:50  a.m.  at  S  in  longitude  48°  26'  west, 
what  is  the  time  at  T  in  longitude  14°  34'  east  ? 

The  employment  of  a  diagram  will  be  of  assistance  in  the  solution 
of  problems  of  this  kind. 

Time  difference  ? 

9:50  A.M.  Time  ? 

8                                      T 
West j j j East 

48°  26'  0°  14°  34' 

Longitude  difference  63°. 

On  a  line  representing  the  equator  locate  the  prime  meridian  0°  and 
,the  meridian  of  each  of  the  places,  writing  under  each  its  longitude 
and  above  >S'  its  time. 


154  Arithmetic 

To  ascertain  the  time  at  T,  we  must  add  to  the  time  at  S  the  time 
difference  between  the  places.  This  is  obtained  from  the  longitude 
difference,  which  is  found  to  be  63°,  48°  26'  +  14°  34'. 

The  time  difference  is  63  hr.  -^  15,  or  4  hr.  12  min.  Adding  9  hr. 
50  min.  and  4  hr.  12  min.,  we  have  14  hr.  2  min.  The  time  at  T 
is,  therefore,  2  min.  past  2  p.m. 

NoTK.  Difference  in  longitude  is  found  by  adding  the  two  longi- 
tudes when  one  is  east  and  the  other  is  west.  The  time  difference  is 
added  to  the  time  of  the  western  place  to  obtain  the  time  of  the  more 
easterly  place. 

11.  When  it  is  1:20  p.m.  at  a  place  in  longitude  27°  15' 
east,  it  is  9:45  a.m.  at  another  place.  Find  the  longitude  of 
the  latter. 

Time  difference  3  hr.  35  min. 
9:45  a.m.  1:20  p.m. 

Of  rp 

West j — — j j East 

Longitude  ?     west     0°  27°  15'  east 

Longitude  difference  ? 

Since  the  time  of  S  is  earlier  than  that  of  T,  the  former  is  placed 
west  of  the  latter,  the  time  difference  being  3  hr.  35  min.  The  longi- 
tude difference  is  (3° 35'  x  15)  53°  45';  that  is,  S  is  53°  45'  west  of  T. 
As  the  latter  is  27°  15'  east  of  the  prime  meridian,  S  must  be  26°  30' 
beyond,  or  26°  30'  west. 

Find  the  longitude  or  the  time : 

Longitude  of  Q.        Longitude  of  li.     .     Time  at  Q.  Time  at  R. 

12.  45°  east  45°  west  9:30  a.m.  ? 

13.  12°  west  154°  west  ?  1:20  a.m. 

14.  27°  30'  east  ?  4:15  p.m.       11:30  a.m. 

15.  ?  78°  15'  east       2:30  p.m.        8  p.m. 

16.  30°  east  ?  10  a.m.  2  p.m. 

17.  47°  20'  west      15°  40'  east  ?  3:30  p.m. 

18.  90°  west  50°  west  1:30  p.m.  ? 

19.  ?  87°  30'  east     12  noon  7:30  a.m. 
ao.    45°  30'  west               ?                 9  a.m.  2:40  p.m. 


Measurements  155 

MEASUREMENTS. 
AREA  OF  RECTANGLES. 

299.  Square  Measure. 

144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.) 

9  square  feet  =  1  square  yard  (sq.  yd.) 

SO^  square  yards  =  1  square  rod  (sq.  rd.) 

160  square  rods  =  1  acre  (A.) 

300.  The  number  of  square  inches  (feet,  yards,  etc.)  in  a 
rectangle  equals  the  product  of  the  nuniher  of  inches  {feet, 
yards,  etc.)  in  its  length  by  the  number  in  its  width. 

301.  Oral  Exercises. 

Find  the  area  of  the  following  rectangles  in  square  inches  : 
Note.     Change  each  dimension  to  inches. 

1.  30  in.  by  20  in.  6.    10  ft.  by  120  in. 

2.  42  in.  by  20  in.  7.    1  yd.  by  20  in. 

3.  48  in.  by  25  in.  8.    1^  ft.  by  10  in. 

4.  1  ft.  by  12  in.  9.    2  ft.  by  30  in. 

5.  1  ft.  by  1  ft.  10.    2  ft.  1  in.  by  1  yd. 

Find  the  area  in  square  feet : 
Note.     Change  each  dimension  to  feet. 

11.  3  ft.  by  3  ft.            '  16.  42  in.  by  24  in. 

12.  36  in.  by  36  in.  17.  30  ft.  by  20  ft. 

13.  1  yd.  by  1  yd.  18.  84  ft.  by  50  ft. 

14.  3  ft.  6  in.  by  2  ft.  19.  12  yd.  by  25  ft. 

15.  3  ft.  6  in.  by  24  in.  20.  10  yd.  by  10  yd. 

Find  the  area  in  square  yards : 
Note.     Change  each  dimension  to  yards. 

21.  3  ft.  by  3  ft.  24.   12  ft.  by  18  ft. 

22.  36  in.  by  36  in.  25.    18  in.  by  72  in. 

23.  li  ft.  by  6  ft.  26.   4  yd.  by  24  ft. 


156  Arithmetic 

27.  18  ft.  by  15  ft.  29.   12  yd.  by  15  ft. 

28.  6  yd.  by  7  yd.  30.    30  ft.  by  30  ft. 

Find  the  area  in  square  rods : 
Note.     Change  each  dimension  to  rods. 

31.  20  rd.  by  15  rd.  36.  240  rd.  by  20  rd. 

32.  55  yd.  by  10  rd.  37.  160  rd.  by  87i  rd. 

33.  165  ft.  by  55  yd.  38.  125  rd.  by  50  rd. 

34.  40  rd.  by  40  rd.  39.  90  rd.  by  30  rd. 

35.  168  rd.  by  50  rd.  40.  120  rd.  by  120  rd. 

Find  the  area  in  acres : 

Note.     Divide  area  in  square  rods  by  160. 

41.  40  rd.  by  4  rd.  46.  80  rd.  by  40  rd. 

42.  40  rd.  by  20  rd.  47.  160  rd.  by  110  rd. 

43.  40  rd.  by  40  rd.  48.  80  rd.  by  104  rd. 

44.  20  rd.  by  20  rd.  49.  124  rd.  by  40  rd. 

45.  80  rd.  by  80  rd.  50.  80  rd.  by  36  rd. 

302.    Oral  Problems. 

1.  How  many  square  inches  in  the  top  of  a  paving 
brick  8  in.  by  4  in.  by  2i  in.?    In  each  side?    In  each  end? 

2.  How  many  square  feet  of  sidewalk  will  144  bricks 
cover,  if  each  covers  a  surface  of  8  inches  by  4  inches  ? 

3.  Find  the  number  of  square  feet  of  pavement  that  will 
be  covered  by  144  bricks  laid  on  the  side  8  inches  by  2^ 
inches. 

4.  How  many  shingles  will  be  needed  to  cover  100 
square  feet  of  roof,  if  each  shingle  covers  a  surface  of  4 
inches  by  4  inches  ? 

5.  How  many  bricks  will  be  required  to  make  100 
square  yards  of  pavement,  if  each  brick  covers  a  surface  of 
8  inches  by  21  inches  ? 


Measurements  157 

6.  The  sides  of  a  house  are  covered  with  shingles,  each 
of  which  covers  a  surface  4  inches  by  6  inches.  How  many 
shingles  are  required  to  the  square  foot  ? 

7.  How  many  boards  12  feet  long  and  1  foot  wide  will 
be  required  for  a  walk  120  feet  long,  6  feet  wide  ? 

8.  How  many  boards  12  feet  long  and  8  inches  wide 
will  be  required  for  a  walk  120  feet  long,  6  feet  wide  ? 

9.  At  $  80  per  acre,  find  the  cost  of  a  square  field,  each 
side  of  which  measures  40  rods. 

10.  Find  the  number  of  square  yards  in  the  ceiling  of  a 
room  18  feet  long,  15  feet  wide. 

11.  How  many  square  yards  in  each  side  wall  of  a  room 
18  feet  long,  9  feet  high  ? 

12.  Find  the  area  in  square  yards  of  each  end  wall  of  a 
room  15  feet  wide,  9  feet  high. 

13.  How  many  square  yards  of  carpet  are  there  in  a  strip 
36  yards  long,  27  inches  wide  ? 

303.    Written  Problems. 

1.  How  many  acres  in  a  rectangular  piece  of   ground 

45  rods  long  and  24  rods  wide  ? 

.         .  45  X  24 

Area  m  acres  = 

160 

2.  How  many  square  yards  can  be  laid  with  900  tiles, 
each  6  inches  square  ? 

3.  A  room  is  18  feet  long,  15  feet  wide,  and  9  feet  high. 
How  many  square  yards  in  each  side  wall?  In  each  end 
wall  ?     In  the  ceiling  ?     In  the  four  walls  and  the  ceiling  ? 

4.  How  many  square  yards  of  carpet  will  be  required 
for  the  floor  of  a  room  18  feet  long,  15  feet  wide  ?  If  the 
carpet  is  1  yard  wide,  how  many  running  yards  are  needed? 
How  many  yards  are  needed,  if  the  carpet  is  f  yd.  wide  ? 

5.  How   many   paving  bricks  will  be   required   for   a 


158  Arithmetic 

sidewalk  300  feet  long  and  12  feet  wide,  the  bricks  being 
laid  on  the  broadest  face  ? 

6.  A  street  300  feet  long  and  36  feet  wide  is  paved  with 
bricks  laid  on  the  side.     How  many  are  needed  ? 

7.  At  15  laths  to  the  square  yard,  how  many  will  be 
used  for  the  walls  and  the  ceiling  of  a  room  18  feet  long, 
15  feet  wide,  9  feet  high,  allowing  for  two  windows,  each 
6  feet  by  4^-  feet,  and  a  door  8  feet  by  4i  feet  ? 

8.  How  many  boards  16  feet  long  and  8  inches  wide 
will  be  needed  for  the  floor  of  a  dock  120  feet  long  and  24 
feet  wide? 

9.  The  owner  of  an  inclosed  lot  100  feet  square  has 
sodded  it,  excepting  a  6-foot  walk  adjoining  the  fence. 
How  many  square  feet  of  sods  will  be  required  ?  How 
many  square  feet  of  paving  will  be  needed  for  the  walk  ? 
How  many  square  feet  of  paving  would  be  required  for  a 
sidewalk  adjoining  the  fence  on  the  outside? 

VOLUME   OF   RECTANGULAR   SOLIDS. 

304.  Cubic  Measure. 

1728  cubic  inches  =  1  cubic  foot  (cu.  ft.) 
27  cu.  ft.  =  1  cubic  yard  (cu.  yd.) 

128  cu.  ft.  =  1  cord  of  wood 

231  cu.  in.  =  1  gallon 

2150.4  cu.  in.  =  1  bushel 

1  cu.  ft.  of  water  weighs  1000  oz. 

305.  Oral  Problems. 

1.  At  71  gallons  to  the  cubic  foot,  what  is  the  capacity 
of  a  cubical  tank  measuring  2  feet  on  a  side  ? 

2.  At  I  bushel  to  the  cubic  foot,  find  the  capacity  of  a 
cubical  bin,  each  side  of  which  measures  10  feet. 

3.  How  many  cubic  inches  in  a  brick  measuring  8  inches 
by  4  inches  by  2^  inches  ? 


Measurements  159 

4.  What  fraction  of  a  cubic  foot  is  the  volume  of  a  brick 
measuring  8  inches  by  4  inches  by  2^  inches  ? 

5.  How  many  cubic  feet  of  air  ape  there  in  a  room  20 
feet  long,  15  feet  wide,  9  feet  high  ? 

6.  How  many  cubic  yards  of  broken  stone  will  be 
needed  to  cover  a  road  300  feet  long,  30  feet  wide,  to  the 
depth  of  18  inches  ? 

7.  If  a  cubic  foot  of  water  weighs  1000  oz.,  how  many 
ounces  does  a  gallon  weigh,  assuming  7^  gallons  to  the  cubic 
foot? 

8.  Assuming  that  zinc  is  8  times  as  heavy  as  water,  how 
many  pounds  will  a  cubic  foot  of  zinc  weigh  ? 

9.  How  many  cubic  yards  of  earth  will  be  removed  in 
digging  a  ditch  300  feet  long,  6  feet  wide,  4i  feet  deep  ? 

10.    How  many  cords  of  wood  are  there  in  a  pile  80  feet 
long,  4  feet  wide,  4  feet  high  ? 

306.    Written  Problems. 

Note.     150'  is  read  150  feet,  18"  is  read  18  inches ;  8"  x  4"  x  2"  is 
read  8  inches  by  4  inches  by  2  inches. 

1.  Find  the  number  of  cubic  yards  in  a  stone  wall  150' 
long,  6'  high,  18"  thick. 

2.  How  many  perches  of  stone,  of  24J  cu.  ft.  each,  are 
there  in  a  wall  8  rods  long,  -6  feet  high,  and  3  feet  thick  ? 

3.  How  many  bricks  8"  X  4"  x  2i"  will  be  required  for 
a  wall  300'  long,  6'  high,  and  18"  thick,  no  allowance  being 
made  for  mortar  ? 

Number  of  bricks  =  ^^^QQ(")x^^(")x^^^^^^.     Cancel. 
8(")  x4(")  x2i(") 

4.  How  many  bricks  will  be  contained  in  each  side  wall 
of  a  house  60  feet  deep  and  40  feet  high,  the  wall  being  12 
inches  thick,  assuming  that  22  bricks  with  mortar  will  make 
a  cubic  foot  ? 


i6o  Arithmetic 

5.  A  bin  is  9  ft.  4  in.  long,  5  ft.  4  in.  wide,  and  2  ft. 
high.  How  many  bushels  of  2150.4  cu.  in.  each  will  it  con- 
tain ? 

6.  Find  the  capacity  in  gallons  (231  cu.  in.)  of  a  tank 
measuring  7  ft.  4  in.  by  4  ft.  8  in.  by  2  ft. 

7.  How  many  cords  are  there  in  a  pile  of  wood  168  ft. 
long,  16  ft.  high,  4  ft.  wide  ?     Find  its  value  at  $4  per  cord. 

8.  A  block  of  dressed  marble  measures  6  ft.  by  4  ft.  by 
2  ft.  What  is  its  weight,  if  marble  is  2.84  times  as  heavy  as 
water  ? 

9.  How  many  cubic  yards  of  earth  will  be  removed 
in  making  an  excavation  24  ft.  wide,  54  ft.  long,  and  12 
ft.  deep?  Find  the  cost  of  hauling  it  at  25^  per  load  of 
a  cubic  yard,  assuming  that  f  cu.  yd.  before  digging  will 
make  1  cu.  yd.  in  the  wagon. 

10.  A  coal  pocket  30  ft.  by  14  ft.  contains  36  tons  of 
coal.  How  high  does  the  coal  stand  in  the  pocket,  if  a  ton 
contains  35  cubic  feet  ? 

BOARD  MEASURE. 

307.  In  the  measurement  of  lumber,  the  unit  is  the  hoard 
foot.  A  board  foot  is  1  ft.  long,  1  ft.  wide,  and  1  in.  thick. 
The  number  of  board  feet  contained  in  a  board  1  in. 
thick  is  equal  to  the  number  of  square  feet  that  it  will  cover. 

308.  To  find  the  number  of  hoard  feet  in  lumber,  multiply 
the  length  in  feet  by  the  width  in  feet  by  the  thickness  in  inches. 

Note.  Boards  less  than  an  inch  thick  are  considered  as  having 
a  thickness  of  an  inch. 

309.  Oral  Exercises. 

Find  the  number  of  board  feet  in  each  of  the  following 
planks,  joists,  sills,  etc.: 

1.  16'  long,  6"  wide,  6"  thick. 

2.  16'  long,  6"  wide,  |"  thick. 


Measurements  i6i 

3.  16'  long,  6"  wide,  i"  thick. 

4.  16'  long,  9"  wide,  2"  thick. 

5.  16'  long,  12"  wide,  3"  thick. 

6.  18'  long,  3"  wide,  2"  thick. 

7.  14'  long,  6"  wide,  2"  thick. 

8.  12'  long,  8"  wide,  4"  thick. 

9.  12'  long,  9"  wide,  2"  thick. 
10.  12'  long,  8"  wide,  3''  thick. 

310.    Written  Exercises. 

1.  Find  the  cost  of  120  boards  16'  long,  10"  wide,  1" 
thick  at  $  35  per  M. 

2.  How  many  board  feet  in  20  3-inch  planks,  each  15 
ft.  long  and  10  in.  wide  ? 

3.  The  sills  for  a  house  consist  of  160  running  feet  8" 
wide,  8"  thick.     Find  the  cost  at  ^50  per  M. 

4.  Find  the  cost  of  fence  material  needed  to  inclose  a 
field  32  rd.  x  40  rd.  with  a  tight  board  fence  6  ft.  high. 
The  boards,  costing  $  14  per  M,  are  nailed  to  posts  placed 
6  ft.  apart,  which  cost  $8.75  per  hundred. 

5.  What  will  be  the  cost  of  a  fence  6  boards  high  in- 
closing a  garden  512  feet  long  and  256  feet  wide,  the  base 
board  being  12"  wide,  the  top  board  6"  wide,  and  the  others 
8"  wide?  The  posts  8  feet  apart,  cost  15  cents  each,  the 
boards  cost  $  48  per  M,  and  labor,  nails,  etc.,  cost  $25. 

6.  For  the  framework  of  a  building  the  following 
materials  are  needed : 

Sills,        8"  X  3",  120  running  feet. 

Joists,   10"  X  3",  900  running  feet. 

Beams,    8"  x  2",  120  running  feet. 

Posts,      5"  X  5",  120  running  feet. 


1 62  Arithmetic 

Eafters,       6"  x  2",   640  running  feet. 
Sheathing,  3"  x  2",  800  running  feet. 
,  Find  the  cost  at  $  20  per  M. 

7.  Find  the  cost  of  the  boards  required  for  1600  square 
feet  of  flooring  at  $  28  per  M,  adding  1  for  waste  in  match- 
ing. 

8.  At  f  1  per  bundle  of  250  shingles,  what  will  be  the 
cost  of  the  shingles  for  a  double  roof,  each  half  measuring 
50'  by  25',  assuming  that  a  shingle  covers  5"  by  4",  and 
adding  yL  of  the  number  for  waste? 

9.  Find  the  cost  at  15  cents  a  square  foot  of  dressing 
6  stone  steps  on  the  top,  one  side,  and  both  ends,  each  step 
being  8  ft.  by  2  ft.  by  3  ft. 

10.  How  many  board  feet  will  be  contained  in  6  wooden 
pillars  9  feet  high  and  18  inches  square  ? 

PRACTICAL   APPLICATIONS. 
MASONRY   AND  BRICKWORK. 

311.  In  calculating  the  cubical  contents  of  walls  for  the 
purpose  of  ascertaining  the  quantity  of  materials  needed, 
openings  are  deducted  and  allowance  is  made  for  mortar. 

312.  When  masons  contract  to  build  walls  by  the  cubic 
foot  or  the  perch,  it  is  their  practice  to  calculate  the  contents 
of  the  walls  inclosing  a  cellar,  for  instance,  by  measuring 
only  the  outside,  the  extra  number  of  cubic  yards  being  con- 
sidered a  compensation  for  the  additional  time  spent  on  the 
corners.  For  openings  in  a  wall,  the  mason  usually  deducts 
one  half  of  the  space ;  this  allowance  for  openings  should 
be  fixed  in  the  agreement. 

313.  Written  Exercises. 

1.  Find  the  number  of  square  feet  of  ground  covered  by 
a  two-foot  wall,  surrounding  a  plot  26  ft.  by  16  ft. 


Measurements 


163 


20" 


16' 


The  outside  measurements  of  the  plot  are  30  ft.  (26  ft.  4-  2  ft.  +  2 
ft.)  by  20  ft.  (16  ft.  +  2  ft.  +  2  ft.).  This 
makes  the  area  of  the  whole  plot  600  square 
feet.  The  area  of  the  inside  plot  is  (26  x  16) 
sq.  ft.,  or  416  sq.  ft.  The  area  covered  by 
the  wall  is  600  sq.  ft.  -  416  sq.  ft.,  or  184 
sq.  ft. 

The  area  of  the  ground  covered  by  the  wall 
may  also  be  found  as  follows  : 

2  sides,  each  .30  ft.  long,  2  ft.  wide     120  sq.  ft. 
2  sides,  each  16  ft.  long,  2  ft.  wide      64  sq.  ft. 

Total 


30  i 


^26- 


16 


I. 

i 


20' 


i^ 


184  sq.  ft. 

2.  How  many  perches  of  masonry  in  a  wall  9  ft.  high 
and  2  ft.  thick  surrounding  a  plot  26  ft.  by  16  ft.  ? 

A  perch  contains  24|  cu.  ft.    Number  of  perches  =  (184  x  9)  -=-  24f . 

3.  The  four  walls  of  a  cellar,  measuring  30  ft.  by  20  ft. 
on  the  outside  of  the  walls,  are  9  ft.  high  and  2  ft.  thick. 
How  many  cubic  feet  of  masonry  will  they  contain,  deducting 
one  half  for  a  door  8'  x  4'  and  4  windows,  each  4'  x  2'  ? 

4.  How  many  cubic  yards  of  stone  in  the  walls  of  a  cellar 
measuring  26'  x  16'  on  the  inside,  the  walls  being  9  ft.  high 
and  18  in.  thick,  and  containing  1  opening  8  ft.  x  4  ft., 
and  4  openings,  each  4  f t.  x  2  ft.,  deducting  \  for  mortar  and 
other  filling  ? 

5.  Find  the  cost,  at  $5  per  perch,  to  build  the  walls  of  a 
cellar  30'  x  20',  outside  measurement,  the  walls  to  be  9  ft. 
high  and  2  ft.  thick,  deducting  one  half  for  an  opening 
8'  X  4',  and  for  four  others,  each  4'  x  2'. 

6.  At  22  bricks  to  the  cubic  foot,  how  many  bricks  will 
be  needed  for  the  walls  of  a  cellar  measuring  30'  x  20'  on 
the  outside,  the  walls  to  be  12"  thick  and  9'  high?  Deduct 
one  half  for  a  door  8'  x  4',  and  for  4  windows,  each  4'  x  2'. 

7.  How  many  cubic  feet  df  concrete  will  be  required  for 
a  floor  4"  thick  in  a  cellar  measuring  28'  by  18'  ?     The  con- 


164  Arithmetic 

Crete  of  this  floor  is  composed  of  a  3-inch  layer,  of  which  ^ 
is  ordinary  cement,  ^  sand,  and  j-  broken  stone.  The  top 
layer,  1  inch,  is  composed  of  equal  parts  of  sand  and  Port- 
land cement.  How  many  cubic  feet  of  each  will  be  required  ? 
8.  How  many  cubic  yards  of  stone  will  be  used  in  building 
a  wall  100  ft.  long,  6  ft.  high,  and  3  ft.  thick,  deducting  i 
for  mortar?  If  the  mortar  is  |  sand  and  i  lime,  how  many 
cubic  yards  of  sand  will  be  needed,  and  how  many  bushels 
of  lime  at  -|  bushel  to  the  cubic  foot  ? 

PAINTING  AND   PLASTERING. 

314.  In  calculating  the  materials  required  in  painting  or 
plastering  a  room,  deduction  is  made  for  all  openings. 

315.  In  ascertaining  the  cost  of  labor,  however,  due  allow- 
ance must  be  made  for  the  extra  time  required  in  working 
on  a  wall  containing  windows,  doors,  and  other  openings. 
For  instance,  a  man  could  kalsomine  an  unbroken  wall  in 
perhaps  less  time  than  he  would  require  to  complete  a  wall 
of  the  same  dimensions  containing  several  windows.  In 
such  a  case,  he  would  be  unwilling  to  make  any  allowance 
for  openings. 

316.  Wooden  laths  are  usually  sold  in  bundles  of  50  or 
100  each.  They  are  4  feet  long  and  generally  1|-  inches 
wide.  Fifteen  laths  of  this  width  will  be  required  for  a 
square  yard,  the  laths  being  laid  about  |  inch  apart  to  per- 
mit the  first  coat  of  plaster  to  pass  between  them  and  thus 
to  obtain  a  better  hold.  Metallic  lathing  of  various  kinds 
is  in  use.  It  is  a  protection  against  fire,  and  its  meshes  in- 
sure a  better  hold  for  the  plaster,  especially  in  ceilings. 

317.  Plastering  is  usually  applied  in  three  coats.  In  the 
first,  called  the  "scratch"  coat,  the  plaster  is  generally 
mixed  with  cattle  hair,  to  render  it  less  liable  to  crumble. 


Measurements  165 

The  second  coat  is  called  the  "  brown  "  coat,  and  the  last  is 
called  the  "  white "  coat.  Patent  plasters  which  require 
only  two  coats  and  need  no  hair,  are  sometimes  employed. 

318.  Outside  painting  requires  at  least  two  coats  for 
new  work.  Ready-mixed  paints  of  all  colors  and  of  all 
qualities  are  easily  procurable.  The  owner  of  a  building 
needs  to  calculate  only  the  quantity  required. 

319.  Written  Exercises. 

1.  How  many  bundles  of  100  laths  will  be  required  for 
4  bedrooms,  each  18'  x  12',  and  9'  high,  deducting  56  sq.  ft. 
for  openings  in  each  room,  and  allowing  18  laths  to  the 
square  yard,  including  w^aste  ? 

2.  Find  the  cost  of  th^  laths  at  40^  per  bundle,  and  the 
cost  of  the  labor  at  6  j^  per  square  yard,  deducting  only  one 
half  for  the  openings. 

3.  How  much  will  be  paid  for  plastering  these  rooms  at 
30  f^  per  square  yard  for  the  material,  and  an  average  of 
5^  per  square  yard  for  each  of  three  coats,  making  full 
deduction  for  openings  in  ascertaining  cost  of  material  and 
a  deduction  of  one  half  in  determining  the  sum  paid  for 
labor  ? 

4.  Wh^t  does  a  contractor  receive  at  45  ^  per  square 
yard  for  plastering  a  room  22  ft.  long,  16  ft.  wide,  and  12 
ft.  high,  allowing  one  half  for  2  windows,  each  6'  x  3|', 
and  a  door  8' X  4' 3"? 

5.  How  much  does  he  pay  for  labor  at  $  4  per  day,  the 
workman  putting  on  132  sq.  yd.  of  the  first  coat  per  day, 
i-  as  much  of  the  second  coat  in  a  day,  and  -J-  as  much  of 
the  finishing  coat  in  a  day  ?     Full  allowance  for  openings. 

6.  What  will  be  the  total  cost  of  painting  the  floor  of 
the  foregoing  room  at  27  cents  per  square  yard,  and  of  kalso- 
mining  the  four  walls  and  the  ceiling  at  18  cents  per  square 
yard,  no  allowance  being  made  for  openings  ? 


1 66  Arithmetic 

7.  Find  the  cost  of  painting  the  outside  of  a  house  at 
27^  per  square  yard,  the  dimensions  being  as  follows: 
each  side  48  ft.  long,  36  ft.  high ;  the  front  and  the  back 

24  ft.  wide  and  24  ft.  high  to  the  eaves 
>Ty  at  C  and  D,  and  36  ft.  high  to  the  ridge 

X  [  \  at  A. 

i' X^)         One  half   allowance  is  made  for   12 

^  windows,   each   6'  x  3i',    and    3    doors, 

each  8'  x  4i'. 

The  area  oi  ACD  =  I  {CD  y.  AB). 

8.  What  will  it  cost  to  stain  the  roof  at  18  cents  per 
square  yard,  each  half  of  the  roof  measuring  21  feet  by  51 
feet  ? 

9.  How  many  gallons  of  paint  will  be  required  to  give 
two  coats  to  the  house,  assuming  that  a  gallon  will  cover 
60  sq.  ft.  the  first  coat  or  80  sq.  ft.  the  second  coat? 
Make  full  allowance  for  openings. 

ROOFING  AND   FLOORING. 

320.  Shingles  are  generally  16  inches  long  and  of  various 
widths.  They  are  j^acked  in  bundles,  each  of  which  equals 
250  shingles  4  inches  wide ;  that  is,  the  total  width  of  the 
shingles  in  a  bundle  is  1000  inches.  The  surface  covered 
by  a  bundle  of  shingles  depends  upon  the  surface  laid  "to 
the  weather."  If  each  shingle  overlaps  12  inches  of  the 
one  beneath,  the  shingles  are  said  to  be  laid  4  inches  to  the 
weather,  and  a  bundle  will  cover  4000  square  inches,  making 
no  allowance  for  waste.  Shingles  are  sold  by  the  thousand, 
consisting  of  four  bundles. 

321.  An  average  roofing  slate  measures  18' inches  by  9 
inches.  Slates  are  generally  laid  7  inches  "  to  the  weather," 
the  surface  then  covered  by  a  slate  being  9"  x  7",  or  63 
square  inches.  If  they  are  laid  6  inches  to  the  weather, 
the  surface  covered  by  a  slate  will  be  54  square  inches. 


Measurements  167 

322.  The  unit  of  measure  in  rooting  is  the  square  of  100 
sq.  ft. 

323.  In  laying  floors,  the  boards  are  "tongued  and 
grooved,"  the  tongue  of  one  board  being  fitted  into  the 
groove  of  the  next.  The  tongue  generally  measures  ^  inch, 
so  that  a  board  3  inches  wide  will  cover  only  2^  inches  of 
the  width  of  the  floor. 

324.  The  number  of  board  feet  required  to  cover  a  sur- 
face with  ordinary  boards  of  1  inch  or  less  in  thickness  is 
equal  to  the  number  of  square  feet  in  the  floor.  In  buying 
tongued  and  grooved  boards,  a  fraction  of  the  number  of 
square  feet  must  be  added  to  find  the  number  of  board  feet 
required. 

Assuming  12  ft.  as  the  length  of  a  board,  and  3  in.  as  its  width  be- 
fore tonguing,  the  number  of  board  feet  is  3.  The  number  of  square 
feet  covered  by  each  is  (12  x  2^)  h-  12,  or  2\ ;  that  is,  to  cover  2\ 
sq.  ft.,  3  sq.  ft.  must  be  paid  for,  or  to  cover  5  sq.  ft.,  6  sq.  ft.  must  be 
paid  for.  The  number  of  square  feet  to  be  paid  for  is  therefore  1^ 
times  the  number  of  square  feet  to  be  covered. 

As  6  sq.  ft.  of  tongued  1-in.  board,  3  in.  wide,  will  cover  only  5 
sq.  ft.  of  floor,  such  boards  cover  only  f  of  the  number  of  square  feet 
(or  board  feet)  purchased. 

325.  Written  Exercises. 

1.  How  many  boards  21  in.  wide  and  12  ft.  long  will  be 
required  for  a  floor  24  ft.  long  and  20  ft.  wide  ?  If  each 
board  has  to  be  paid  for  as  being  3  in.  wide,  measuring  the 
tongue,  how  many  board  feet  of  floor  must  be  ordered  ? 

Area  to  be  covered,  480  sq.  ft. 
Area  covered  by  1  board,  2^  sq.  ft. 
Number  of  boards,  480  -^  2^. 

2.  How  many  square  feet  of  surface  will  be  covered  by 
192  boards  12  feet  long,  3  inches  wide,  if  \  of  the  surface 
of  the  boards  is  deducted  for  tonguing  ? 


1 68  Arithmetic 

3.  Find  the  number  of  board  feet  of  flooring  that  is 
required  for  a  room  32'  x  27^',  the  board  being  3  inches 
wide,  including  i  inch  of  tonguing. 

The  number  of  square  feet  to  be  covered  =  32  x  27|. 

Width  of  board  less  tonguing  =  2^"  =  -^-^ . 

Total  length  of  boards  required  =  (32  x  27^)  -4-  ^\. 

Find  the  number  of  board  feet  in  boards  of  the  required  length  and 
3  inches  wide,  or  \  foot. 

Compare  the  answer  thus  obtained  with  the  result  obtained  by 
adding  to  the  area  of  the  floor  \  of  the  area. 

4.  Find  the  number  of  board  feet  of  flooring  that  is 
required  for  a  room  32'  x  27^',  the  boards  to  be  6  inches 
wide,  including  \  inch  of  tonguing. 

Note.  Why  does  this  result  differ  from  the  result  in  the  preceding 
problem  ? 

5.  What  fraction  of  a  3-inch  board  is  lost  in  the  tongu- 
ing ?     What  fraction  of  a  6-inch  board  is  lost  ? 

What  fraction  of  the  surface  of  a  floor  must  be  added 
when  3-inch  boards  are  used  ?  What  fraction  when  6-inch 
boards  are  used  ? 

6.  How  many  bundles  of  shingles  laid  4  inches  to  the 
weather  will  be  required  to  cover  a  roof,  each  half  of  which 
measures  50  feet  by  20  feet  ? 

7.  How  many  days  will  a  man  take  to  shingle  a  roof 
50  feet  by  40  feet,  if  he  lays  2000  shingles  per  day  ? 

8.  Find  the  cost  of  slating  a  roof  56'  x  45'  at  $16  per 
square  (100  sq.  ft.). 

9.  How  many  slates  are  required,  if  each  slate  covers 
9"  X  7"  ?  How  many  square  feet  of  slates  are  used,  each 
slate  measuring  18"  x  9"  ? 

CARPETING  AND  PAPERING. 

326.  Ingrain  carpet  and  matting  are  generally  1  yard 
wide.     Other  carpets  are  J  yard,  or  27  inches,  wide.     Lino- 


Measurements  169 

lenm  and  oilcloth  are  made  in  various  widths.     All  are  sold 
by  the  running,  or  linear,  yard. 

327.  Wall  paper  is  sold  in  rolls.  It  is  usually  18  inches 
wide  and  the  single  rolls  are  8  yards  long,  the  double  rolls 
being  16  yards  long. 

328.  Owing  to  the  loss  in  matching  patterns,  it  is  diffi- 
cult to  ascertain  the  exact  quantity  of  carpet  required  for  a 
given  room. 

The  difficulty  is  increased  in  the  case  of  wall  paper,  the 
location  of  windows,  doors,  and  other  openings,  frequently 
necessitating  much  waste. 

329.  Written  Exercises. 

1.  How  many  yards  of  carpeting  1  yd.  wide  will  be 
required  for  the  floor  of  a  room  24  ft.  long,  18  ft.  wide? 

If  the  pattern  is  repeated  at  intervals  of  any  aliquot  part  of  a  yard, 
there  will  be  no  loss  in  matching.  As  the  room  is  8  yd.  long  and  6 
yd.  wide,  it  will  require  8  strips  of  carpet,  each  containing  6  yd.,  or 
6  strips,  each  containing  8  yd.  Ans.  48  yd. 

2.  How  many  yards  of  carpeting  f  yd.  wide  will  be 
required  for  the  floor  of  a  room  24  ft.  long,  18  ft.  wide? 

If  the  strips  run  the  length  of  the  room,  each  strip  will  be  8  yd. 
long.  The  number  of  strips  is  6  yd.  h-  |  yd.,  or  8.  The  number  of 
linear  yards  is  8  x  8,  or  64.  Ans.  64  yd. 

3.  How  many  yards  of  carpeting  f  yd.  wide  will  be  re- 
quired for  the  floor  of  a  room  25  ft.  long,  19  ft.  wide  ? 

If  the  strips  run  the  length  of  the  room,  each  strip  will  be  8^  yd. 
long.  Dividing  19  ft.  by  2^  ft.,  we  get  a  quotient  of  8,  and  a  re- 
mainder of  1  ft.  It  is  necessary  to  buy  a  ninth  strip  to  obtain  the 
piece  1  ft.  wide  necessary  to  complete  the  carpet.  8i  yd.  x  9  =  75 
yd.  Ans.  75  yd. 

If  the  strips  run  across  the  room,  each  strip  will  be  6^  yd. 
long.  The  number  of  strips  required  will  be  12(25 -=- 2^  =  11+). 
6i  yd.  X  12  =  76yd.  ^ns.  76  yd. 

One  yard  will  be  saved  by  carpeting  the  room  the  short  way. 


lyo  Arithmetic 

4.  How  many  yards  of  carpet  1  yard  wide  will  be  re- 
quired for  the  floor  of  a  room  25  feet  long,  19  feet  wide,  if 
the  strips  run  the  length  of  the  room  ? 

5.  How  many  yards  of  carpet  1  yd.  wide  will  be  needed 
for  the  floor  of  the  same  room,  if  the  strips  run  the  width  of 
the  room  ? 

6.  How  many  rolls  of  paper  8  yards  long,  18  inches  wide 
will  be  required  for  the  ceiling  of  a  room  24  feet  long,  18 
feet  wide  ? 

7.  If  the  foregoing  room  is  9  ft.  high  above  the  base- 
board and  below  the  border,  how  many  single  rolls  of  paper 
will  be  required  for  the  walls,  deducting  for  two  windows, 
each  6  ft.  x  4i  ft.,  and  a  door  8  ft.  x  6  ft.,  no  waste  arising 
in  matching  the  pattern,  etc.  ? 

8.  Find  the  cost  of  papering  the  walls  and  the  ceiling  of 
the  foregoing  room,  at  50  i^  per  roll  for  the  paper,  and  10  ^ 
per  running  yard  for  the  border  around  the  four  walls,  add- 
ing $  3  additional  for  labor,  etc. 

9.  Find  the  cost  of  carpeting  this  room  with  carpet  27 
inches  wide,  costing  %  1.25  per  running  yard,  and  10  cents 
per  square  yard  for  sewing,  lining,  and  laying. 

10.  The  floor  of  a  room  14  ft.  by  18  ft.  is  to  be  covered 
with  matting  1  yd.  wide.  Which  will  be  the  cheaper  way 
to  lay  the  matting,  and  how  much  will  be  saved  if  the 
matting  costs  65  ^  per  yard  ? 

11.  The  floor  of  a  kitchen  24  ft.  by  16^  ft.  is  covered  with 
linoleum  2  yd.  wide.  Find  the  cost  at  $1  per  square  yard, 
the  strips  running  the  more  economical  way. 

12.  A  floor  16'  X  12'  is  to  be  covered  with  carpet  27  in. 
wide,  at  $2  per  yard,  made  and  laid.  What  will  be  the 
cost,  if  9  inches  must  be  cut  from  the  length  of  each  strip, 
except  the  first,  to  match  the  pattern,  the  carpet  running 
the  length  of  the  room  ? 


The  Metric  System  171 

THE   METRIC   SYSTEM. 

330.  The  metric  system  of  weights  and  measures  was 
devised  in  France  about  the  beginning  of  the  last  century. 
It  has  been  adopted  in  many  countries,  and  legalized  in 
others,  including  the  United  States.  In  this  country  its 
use  is  largely  limited  to  the  compounding  of  prescriptions 
by  druggists  and  to  the  scientific  work  of  high  schools  and 
colleges. 

331.  The  countries  that  use  the  metric  system  in  all 
their  transactions  find  it  of  great  advantage  in  their  ex- 
change of  commodities,  not  being  compelled,  as  we  are,  in 
buying  goods  in  France,  for  instance,  to  change  the  meters, 
liters,  and  kilograms,  into  yards,  gallons,  and  pounds,  in 
order  to  compare  prices. 

332.  The  basis  of  the  metric  system  is  the  meter,  which 
is  one  ten-millionth  part  of  the  quarter  meridian  passing 
through  Paris.  The  meter  is  39.37  inches,  or  a  little  over 
IjL  yard. 

333.  The  liter  is  the  unit  of  volume.  It  has  the  capacity 
of  a  cube,  each  edge  of  which  is  a  decimeter,  or  -^  meter. 
It  is  a  little  larger  than  our  liquid  quart. 

334.  The  unit  of  weight  is  the  gram.  A  liter  of  pure 
water  at  a  certain  temperature  weighs  1000  grams,  or  a  kilo- 
gram, about  2\  pounds. 

335.  In  changing  from  one  unit  to  another  of  the  same 
table,  the  reduction  is  effected  by  the  removal  of  the  decimal 
point  or  by  the  annexing  of  ciphers. 

336.  In  naming  the  units  of  this  system  the  following 
prefixes  are  used : 


172  Arithmetic 

dec!  meaning  J^  deka  meaning  10 

centi  meaning  yi^  hecto  meaning  100 

milli  meaning  jiho  ^i^^  meaning  1000 

myria  meaning  10,000 

LONG  MEASURE. 

337.  Table. 

10  millimeters  (""")=  1  centimeter  (^"^) 

10  centimeters  =  1  decimeter  C^"') 

10  decimeters  =  1  meter  ('") 

10  meters  —  1  dekameter  (i^">) 

10  dekameters  =  1  hectometer  ("™) 

10  hectometers  =  1  kilometer  (^"') 

10  kilometers  =  1  myriameter  (Mm^ 

338.  The  measures  in  common  use  are  the  millimetery  the 
centimeter,  the  meter,  and  the  kilometer. 

Our  nickel  5^  piece  is  2  centimeters  in  diameter  and  2  millimeters 
thick.  An  opening  through  which  a  nickel  will  just  pass  is  2  centi- 
meters long  and  2  millimeters  wide.  The  accompanying  scale  is  8 
centimeters  long  and  2  millimeters  wide. 

I  I 1  I  I  I  I  I  I 


339.  The  meter  is  employed  in  denoting  short  distances, 
such  as  we  measure  in  yards.  The  centimeter  is  employed 
to  express  widths  of  ribbon,  dress  goods,  carpets,  boards, 
etc.  The  thickness  of  coins,  the  diameter  of  wire,  etc.,  are 
expressed  in  millimeters.  Long  distances  are  given  in 
kilometers. 

340.  Sight  Exercises. 

1.  Change  3570'"  into  millimeters ;  into  centimeters ;  into 
kilometers. 

2.  How  many  centimeters  in  584'"  ?  How  many  milli- 
meters ?     What  decimal  of  a  kilometer  ? 


The   Metric  System  173 

3.  Add  42"^  and  STS'^™. 

4.  From  30*"  take  250'^"\ 
6.    Multiply  2.6™  by  10. 

6.  Divide  45*"  by  100. 

7.  How  many  nickels  laid  side  by  side  will  measure  a 


meter  ? 

« 

8.  How  many  dollars   are  there  in  a  pile  of  nickels  1 

meter  in  height  ? 

9.  How  many  strips  of  carpet,  each  75  centimeters  wide, 
will  be  required  to  cover  a  floor  9  meters  wide  ? 

10.  If  a  man's  step  is  To*^™,  how  many  steps  will  he  take 
in  going  f  of  a  kilometer  ? 

11.  Assuming  a  kilometer  to  be  f  mile,  how  many  kilo- 
meters will  equal  100  miles  ? 

12.  If  a  boy  wears  a  12-inch  collar,  for  what  size,  in  centi- 
meters, would  he  ask  in  a  store  in  Paris  ? 

13.  What  is  the  approximate  width  in  inches  of  a  board 
25  centimeters  wide  ? 

14.  The  unit  of  thickness  in  French  board  measure  is  25 
millimeters.  What  is  the  corresponding  thickness  in  inches, 
assuming  the  meter  to  be  40  inches  ? 

15.  Assuming  the  meter  as  3  ft.  3  in.,  how  many  feet  in 
length  is  a  board  4  meters  long  ? 

16.  Two  towns  are  120  kilometers  apart.  What  is  the  dis- 
tance between  them  in  miles  at  |  mile  to  the  kilometer  ? 

341.    Written  Exercises. 

1.  Express  the  sum  of  the  following  in  meters : 

3.46'"  +  2^""  -f  415*='"  -f  18""". 

2.  From  1  kilometer  take  295  centimeters.     Give  result 
in  meters. 


174  Arithmetic 

3.  How  many  meters  in  75  hundredths  of  18.48  kilo- 
meters ? 

4.  Give  in  centimeters  the  quotient  of  84*"  -^  .75. 

5.  How  many  strips  of  matting,  each  95  centimeters 
wide,  will  be  required  for  the  floor  of  a  room  11.4  meters 
wide  ?  If  the  room  is  13.5  meters  long,  how  many  linear 
meters  of  matting  will  it  take  to  cover  the  floor  ? 

6.  Find  the  total  cost  of  the  following : 

2.30  meters  of  cloth  @  9.50  francs. 
2.50  meters  of  satin  @  6.40  francs. 
4.75  meters  of  ribbon  @  3.50  francs. 

7.  To  make  a  dozen  napkins,  10.80  meters  of  linen  are 
required.  How  many  meters  are  necessary  to  make  125 
napkins  ? 

8.  A  dealer  pays  1836  francs  for  3  pieces  of  cloth  cost- 
ing 8.50  francs  per  meter.    How  many  meters  in  each  piece  ? 

9.  A  man  receives  4.45  francs  per  day.  How  many 
francs  will  he  receive  for  digging  a  ditch  63.45  meters  long, 
if  he  digs  2.35  meters  in  a  day  ? 

10.    A  man  takes  120  steps  of  70  centimeters  each  in  a 
minute.     How  many  meters  will  he  travel  in  an  hour  ? 

SQUARE   MEASURE. 

342.    Preliminary  Exercises. 

1.  How  many  square  decimeters  in  a  square  10  deci- 
meters long,  10  decimeters  wide  ? 

2.  How  many  square  meters  in  a  square  1  meter  long, 
1  meter  wide  ? 

3.  How  many  square  decimeters  in  a  square  meter  ? 

4.  How  many  square  centimeters  in  a  square  1  deci- 
meter long  and  1  decimeter  wide  ? 

5.  How  many  square  centimeters  in  a  square  decimeter  ? 


The   Metric  System  175 

343.  Table. 

100  sq.  centimeters  (<i'^"i)  =  1  sq.  decimeter  (qdm) 

100  sq.  decimeters  =  1  sq.  meter  (q"^) 

100  sq.  meters  =  1  sq.  dekameter  (qOm) 

100  sq.  dekameters  =  1  sq.  hektometer 

100  sq.  hektometers  =  1  sq.  kilometer 

344.  In  calculating  areas,  express  the  length  and  the  tvidth 
in  the  same  linear  unit.  The  product  of  these  ttvo  mimhers 
will  give  the  area  in  square  units  of  the  same  name. 

345.  Sight  Exercises. 

1.  How  many  sq.  decimeters  in  a  square  11  decimeters 
long,  11  decimeters  wide  ? 

2.  How  many  sq.  meters  in  a  square  1.1'"  X  1.1™? 

3.  Change  121  sq.  decimeters  to  sq.  meters.     How  many 
decimal  places  are  pointed  off  ? 

4.  Change  121  decimeters  to  meters.    How  many  deci- 
mal places  are  pointed  off  ? 

5.  How  many  square  meters  in  a  rectangle  44'"  x  25"  ? 

6.  How  many  square  dekameters  in  a  rectangle  4.4^"" 
X  2.5^'"  ? 

7.  Change  1100  square  meters  to  square  dekameters. 

8.  Change  11  square  dekameters  to  square  meters. 

9.  How  many  square  meters   in   the   floor   of   a    room 
8  meters  long,  5.5  meters  wide  ? 

10.    How  many  square  meters  in  the  surface  of  a  board 
4  meters  long,  25  centimeters  wide  ? 

DRY  AND   LIQUID   MEASURE. 

346.  Table. 

10  deciliters  (d»)  =  1  liter  (i) 

10  liters  =  1  dekaliter  (DI) 

10  dekaliters  =  1  hektoliter  (Hi) 


176  Arithmetic 

347.  A  liter  is  the  equivalent  of  a  hollow  cube  1  deci- 
meter long,  1  decimeter  wide,  1  decimeter  high.  The  liter 
is  the  unit  employed  in  buying  and  selling  all  liquids  and 
grains  that  are  measured.  Large  quantities  are  expressed 
in  hektoliters.  All  the  other  prefixes  can  be  employed,  but 
those  given  in  the  table  are  the  most  common. 

348.  To  ascertain  the  contents  of  a  rectangular  vessel  in 
liters,  find  the  product  of  the  three  dimensions,  each  expressed 
in  decimeters. 

349.  Written  Exercises. 

1.  How  many  hektoliters  of  grain  will  a  bin  contain 
40™  long,  2"^  wide,  1.2""  deep  ? 

2.  A  tank  measures  4"'  x  2""  x  1.2™.  How  many  liters 
of  oil  will  it  hold  ? 

3.  A  piece  of  ground  20™  x  10™  produced  5^^  2^  of  beans. 
What  is  the  yield  of  a  square  meter  in  deciliters  ? 

4.  A  field  produces  648  sheaves  of  wheat,  4  sheaves 
yielding  a  double  dekaliter  of  grain.  How  many  hektoli- 
ters in  the  crop  ? 

5.  A  pipe  discharges  28  liters  of  water  in  5  seconds. 
How  many  hektoliters  are  discharged  in  24  hours  ? 

350.  Table  of  Weight. 

10  milligrams  ('"s  )  =  1  centigram 

10  centigrams  =  1  decigram 

10  decigrams  =  1  gram  (») 

10  grams  =  1  dekagram 

10  dekagrams  =  1  hektogram 

10  hektograms  =  1  kilogram  C^s) 

1000  kilos  =  1  metric  ton  (t) 

351.  A  cubic  centimeter  of  water  weighs  1  gram.  A 
liter  of  water  weighs  1000  grams,  called  a  kilogram  or  kilo. 
A  cubic  meter  of  water  weighs  a  metric  ton. 


The   Metric  System  177 

352.  Sight  Exercises. 

1.  A  vessel  measures  3*^™  X  3*^"*  x  3*^™.  How  many  liters 
will  it  contain  ? 

2.  Find  the  weight  in  kilos  of  the  water  that  can  be  con- 
tained in  a  cubical  measure  S*^""  x  S*^'"  x  3"^"'. 

3.  If  cast  iron  is  7.5  times  as  heavy  as  water,  how  many 
kilos  will  an  iron  cube  weigh,  each  side  of  which  measures 

4.  Find  the  weight  in  metric  tons  of  the  material  re- 
moved in  excavating  a  cellar  8™  x  o""  x  3™,  assuming  that 
it  is  twice  as  heavy  as  water. 

5.  Mercury  is  13.6  times  as  heavy  as  water.  How  many 
kilos  does  a  liter  of  mercury  weigh  ? 

6.  How  many  kilos  in  a  cubical  block  of  marble,  each 
side  of  which  measures  1™,  marble  being  2.837  times  as 
heavy  as  water  ? 

7.  An  empty  pail  weighing  2  kilos  is  partly  filled  with 
water.  How  many  liters  of  water  does  it  contain,  if  the 
weight  of  the  pail  and  the  water  is  10  kilos  ? 

8.  A  5^  nickel  piece  weighs  5  grams.  How  many  dol- 
lars in  nickels  will  weigh  a  kilo  ? 

9.  Find  the  weight  of  a  double  dekaliter  of  wheat  at 
75  kilos  per  hektoliter. 

10.  A  franc  contains  100  centimes.  Bronze  coins  of  5 
and  10  centimes  weigh  5  and  10  grams  respectively.  What 
is  the  value  in  francs  of  a  kilo  of  bronze  coins  ? 

353.  Written  Exercises. 

1.  What  is  the  weight  of  the  hay  consumed  by  a  horse 
in  30  days,  if  he  eats  7*^^  5"^  per  day  ? 

2.  What  is  the  price  of  a  liter  of  olive  oil  weighing 
920  grams  at  2^  francs  per  kilo  ? 


lyS  Arithmetic 

3.  The  front  wheels  of  a  carriage  are  3.25™  in  circum- 
ference.    How  many  revolutions  will  they  make  in  going 

4.  A  train  runs  252.9  kilometers  in  5  hours.  What  is  its 
speed  in  meters  per  minute  ? 

5.  An  empty  vase  weighs  2947  grams ;  filled  with  olive 
oil,  it  weighs  15,757  grams.  What  is  the  capacity  of  the 
vase,  if  a  liter  of  oil  weighs  915  grams  ?  How  much  would 
it  weigh  if  filled  with  water  ? 

354.  COMPARISON  OF  UNITS. 

L     oth    1^^^®^'        39.37  in.  Yard,         .9144  meter 

^^^        [Kilometer,     .62137  mi.  Mile,       1.6093  kilometers 

'  liquid  qt.  Liquid  qt.,    .9463  liter 

dry  qt.  Dry  qt.,       1.101  liters 


^         ......      [1.0567  liquid  qt.  Liquid  qt.,    .9463  liter 

capacity    Literj    ^^^  ^^^ 


W  Vht    J^^^'^        15.432  gr.  troy  Troy  gr.      6.48  milligrams 

^^^        [Kilo  2.2046  lb.  avoir.       Avoir,  lb.      .4536  kilo 

355.   Sight  Exercises. 

1.  How  many  meters  in  1000  yd.  ? 

2.  How  many  miles  in  100  kilometers  ? 

3.  How  many  quarts,  liquid  measure,  in  100  liters  ? 

4.  How  many  liters  in  100  quarts,  dry  measure  ? 

5.  How  many  troy  grains  in  a  kilo  ? 

6.  How  many  kilos  in  1000  pounds  avoirdupois  ? 


CHAPTER   VI. 

RATIO  AND  PROPORTION;    POWERS   AND   ROOTS. 
ARITHMETICAL   ANALYSIS. 

356.   Sight  Exercises. 

1.  If  3  sheep  cost  $12f,  what  will  8  cost? 

2.  At  the  rate  of  $4  for  25  pounds  of  cheese,  how  many 
pounds  can  I  buy  for  f  1.76? 

3.  If  7  men  can  do  a  piece  of  work  in  12  days,  how  long 
would  it  take  4  men  to  do  it  ? 

4.  What  will  be  the  cost  of  3\  yards  of  dress  goods  at 
the  rate  of  16  cents  per  half  yard? 

5.  A  garrison  requires  140  barrels  of  flour  every  2 
weeks.     Find  the  time  that  490  barrels  will  last. 

Each  of  the  foregoing  examples  is  most  readily  solved  "mentally" 
by  the  method  of  unitary  analysis,  as  it  is  sometimes  called.  In  No. 
l,.the  cost  of  1  sheep  is  first  found  ;  in  No.  2,  the  cost  of  1  pound ;  in 
No.  3,  the  time  required  by  1  man,  etc. 

In  the  next  five  exercises,  the  ratio  method  is  preferable. 

6.  Find  the  cost  of  33  sheep  at  the  rate  of  $  13  for  3 
sheep,     (^ws.  11  times  $  13.     Why?) 

7.  At  ^22  per  dozen,  what  should  be  paid  for  36  hats? 
(How  many  times  ^22?) 

8.  If  7  men  can  do  a  piece  of  work  in  150  days,  how 
long  would  it  require  21  men  to  do  the  same  work  ?  (What 
fraction  of  150  days?) 

9.  Find  the  cost  of  3  yards  of  merino  at  the  rate  of  22^ 
for  I  yd.     (How  many  times  22  ^  ?) 

10.    If  a  garrison  uses  111  barrels  of  flour  every  4  weeks, 
how  long  will  333  barrels  last? 

179 


1 80  Arithmetic 

11.  When  I  lb.  butter  costs  21  p,  find  the  cost' of  li  lb. 

Oral  problems  involving  the  comparison  of  two  fractional  quanti- 
ties may  be  solved  by  giving  the  fractions  the  same  denominator. 

Since  1^,  or  f,  equals  J/-,  the  foregoing  problem  requires  the  cost 
of  12  eighths  when  7  eighths  cost  21j?^.  The  comparison  may  now  be 
made  between  12  and  7  as  follows,  the  eighths  being  ignored: 

When  7  parts  cost  21^,  1  part  costs  3^,  and  12  parts  cost  36^. 
Ans. 

Solve  the  next  five  problems  by  reducing  the  fractions  to  a  common 
denominator. 

12.  I  pay  $8000  for  2  thirds  of  a  farm.  What  should  I 
pay  for  3  fourths  of  it  at  the  same  rate? 

13.  A  man  sells  a  cow  for  |-  of  its  cost,  receiving  for  it 
$  48.  What  would  he  have  received,  if  he  had  sold  the  cow 
for  f  of  its  cost? 

14.  A  roll  of  matting  f  yd.  wide  contains  27  sq.  yd. 
How  many  square  yards  in  another  roll  of  the  same  length, 
but  11-  yd.  wide? 

15.  If  1^  of  a  number  is  24i,  what  is  -^^  of  the  same  num- 
ber? 

16.  A  person  owning  ^t  of  a  business  receives  $4000  as 
his  share  of  the  profits.  How  much  should  the  owner  of 
-^^2  of  the  business  receive  ? 

17.  At  3  pairs  for  50  cents,  what  will  be  the  cost  of  li 
dozen  pairs? 

18.  A  wagon  load  of  sand  containing  2  cubic  yards  costs 
$l.oO.     What  is  the  cost  of  28  cubic  yards? 

19.  How  long  will  a  supply  of  provisions  last  75  men, 
if  the  same  quantity  will  last  25  men  7^  months? 

20.  A  field  of  grass  can  be  mowed  by  8  men  in  4^  days. 
How  many  men  will  be  required  to  mow  it  in  6  days? 

21.  By  hauling  hay  to  market  in  loads  of  f  ton  each,  72 
trips  are  required.  How  many  trips  will  be  saved,  when  the 
road  is  so  improved  that  a  load  of  l^-  tons  can  be  hauled? 


Arithmetical   Analysis  i8i 

22.  Lumber  for  a  building  ean  be  hauled  in  16  days,  if 
the  teams  haul  3  loads  per  day.  How  many  days  will  be 
required  if  the  teams  haul  4  loads  per  day  ? 

23.  A  man  has  sufficient  hay  to  last  his  8  horses  for  12 
weeks.     How  long  will  it  last  if  he  sells  2  of  his  horses? 

24.  There  are  required  for  a  barn  floor  90  boards  14 
inches  wide.  How  many  will  be  needed,  if  9-inch  boards 
are  used? 

25.  The  captain  of  a  vessel  has  60  days'  provisions  for 
36  passengers.  How  many  additional  passengers  can  he 
take,  if  the  voyage  lasts  but  48  days  ? 

357.   Written  Problems. 

1.  Find  the  cost  of  96  bbl.  flour,  when  34  bbl.  cost 
$  199.75. 

Instead   of  finding  the  cost  of  1  bbl.  by  dividing  $199.75  by  34 

and  multiplying  the  quotient  by  96,  it  will  generally  be  found  more 

satisfactory  to  leave  division  for  the  last       ^.^^  „. 

S199  7o  X  96 
operation.     The  cost  of  1  bbl.  is  indicated  Cancel. 

by  writing  34  as  a  divisor,  and  the  cost 

of  96  is  indicated  by  writing  96  as  a  multiplier. 

2.  When  the  lamps  in  a  factory  burn  2  hr.  40  min.  per 
day,  a  barrel  of  oil  lasts  30  da.  How  long  will  it  last  when 
the  lamps  barn  3  hr.  20  min.  per  day? 

Change  the  time  to  160  min,  and  200  min. 

When  the  lamps  burn  160  min.  per  day,  the  oil  lasts  30  da.;  when 
they  burn  1  min.,  the  oil  will  last  160       39  da   x  160 
times  as  long  (write  160  in  the  numera-  ^^  Cancel. 

tor)  ;  when  they  burn  200  min.  per  day, 

the  oil  will  last  2^  ^^  ^1^®  ^^™^  ^^  would  last  if  the  lamps  burned  1  min. 
per  day.     (Write  200  in  the  denominator.) 

3.  A  farmer  sells  j\  of  his  farm  for  $  2362.50.  What  is 
the  value  of  f  of  the  farm  at  the  same  rate  ? 


1 82  Arithmetic 

In  written  examples  it  is  often  unnecessary  to  change  the  fractions 
to  those  having  a  common  denominator. 

As  j%  of  the  farm  is  worth  a  cer-  $2862.50  x  f 

tain  sum,  the  farm  is  worth  this  sum  ^^ 

divided  by  ^%,  and  f  of  the  farm  is       ^^„, , ,^      ^ 

.,   .u        1         wi     f  u-        $2362.50x16x6      ^o^^^i 

worth  the  value  of  the  farm  multi- Cancel. 

9x7 
plied   by  f.      These  fractions  can 

first  be  written  above  and  below  the  line,  respectively,  in  the  manner 
shown  above,  and  then  rewritten,  -^^  being  inverted,  and  the  denomi- 
nator being  written  above  the  line  and  the  numerator  below. 

4.  If  a  bag  containing  24^  pounds  of  flour  will  make 
29^  pounds  of  bread,  how  many  pounds  of  bread  can  be 
made  from  a  barrel  of  flour,  196  pounds  ? 

5.  If  20°  on  the  equator  measures  1383.2  miles,  what  is 
the  equatorial  circumference  of  the  earth  ? 

6.  A  farmer  has  sufficient  hay  to  last  120  sheep  36 
days.     How  long  would  it  last  96  sheep  ?     160  sheep  ? 

7.  How  many  sheep  could  be  fed  for  32  days  with  a 
supply  of  hay  that  would  last  120  sheep  36  days  ? 

8.  To  empty  a  pond  by  a  pump  discharging  28  gallons 
of  water  per  minute  requires  15  days.  How  long  would  it 
take  a  pump  that  discharges  35  gallons  per  minute? 

9.  A  man  on  a  journey  finds  that  at  $  8  per  day  for  ex- 
penses he  can  travel  only  45  days  longer.  How  many  days 
can  he  prolong  his  trip  by  reducing  his  expenses  by  one 
tenth  ? 

10.  If  a  traveler  has  sufficient  money  to  last  him  110 
days,  how  many  days  will  his  trip  be  shortened  if  he  in- 
creases his  daily  expenses  one  tenth  ? 

11.  A  farmer  has  a  quantity  of  hay  sufficient  to  last  his 
cattle  a  certain  time.  By  what  fraction  is  the  time  dimin- 
ished, if  he  increases  his  herd  -^^  ?  By  what  fraction  is  the 
time  increased,  if  he  diminishes  his  herd  -^^  ? 

12.  If  I  pay  $120  for  the  use  of  a  piece  of  land  for  2  yr. 
6  mo.,  what  should  I  pay  for  its  use  for  3  yr.  4  mo.? 


Men 

Da. 

Hr. 

Ft.  W. 

Ft.  D. 

30 

40 

10 

6 

4 

1 

1 

1 

1 

1 

50 

60 

8 

8 

3 

Arithmetical  Analysis  183 

13.  A  loans  B  a  certain  sum  with  the  understanding  that 
he  is  to  receive  $160  for  its  use  for  3  yr.  4  mo.  If  B 
returns  it  in  2  yr.  6  mo.,  what  should  he  pay  for  its  use  ? 

14.  If  6  men  can  do  a  certain  piece  of  work  in  24  days, 
how  long  will  it  take  4  men  and  4  boys  to  do  it,  if  a  boy 
does  one  half  as  much  work  as  a  man  ? 

This  method  of  solving  problems  by  analysis  is  also  employed  when 
the  problems  contain  a  large  number  of  conditions,  but  involve  only 
multiplication  and  division. 

15.  If  30  men  in  40  da.,  working  10  hr.  a  day,  dig  a  ditch 
450  ft.  long,  6  ft.  wide,  and  4  ft.  deep,  how  long  a  ditch  8  ft. 
wide  and  3  ft.  deep  can  50  men  dig  in  60  da.  working  8 
hr.  a  day  ? 


450  ft.  .  . 

(a) 


Arrange  the  work  as  shown  in  the  first  horizontal  row,  each  number 
having  its  designation  written  above  it.  Write  450  ft.  last,  with  a  line 
drawn  under  it.  In  the  second  row  place  1  in  the  column  of  meji  and 
reason  thus : 

"  If  30  men  dig  450  ft.,  1  man  will  dig  3L  ^^^  ^^  x  6  x  4 

4^0  f^-'"  30  x'40  X  10        (^) 

and  write  30  in  the  denominator.     Next 

place  1  in  the  column  of  days,  and  reason  thus : 

"  Working  1  da.,  he  will  do  ^-^  as  much,  etc.," 

and  write  40  in  the  denominator.  Then  place  1  in  the  column  of 
hours,  and  reason : 

"  Working  1  hr.  per  day,  he  will  do  ^^,  etc.," 

and  write  10  in  the  denominator.  Place  1  in  the  column  of  feet 
loide,  and  reason  : 

"  When  the  ditch  is  1  ft.  wide,  he  can  do  6  times  as  many  feet  of 
length," 

and  write  6  in  the  numerator.  Place  1  in  the  column  of  feet  deep, 
and  reason : 

"When  the  ditch  is  1  ft.  deep,  he  can  do  4  times  as  many  feet 
of  length," 

and  write  4  in  the  numerator. 


184  Arithmetic 


The  next  step  is  to  compare  successively  50  men,  60  da.,  8  hr., 
8  ft.  wide,  3  ft.  deep,  with  1  man,  1  da.,  1  hr.,  1  ft.  wide,  1  ft.  deep. 
Placing  50  in  the  column  of  me«,  reason  : 

"  50  men  will  dig  50  times 

,  ,.,  ,        "1  „  450  ft.  X  6  X  4x  50  X  60  X  8 

as  long  a  ditch  as   1    man," — (q) 

T      .,      .f.   •     ^u  30  X  40  X  10  X  8  X  3  ^""^ 

and  write    50   in  the   nu- 
merator.    Placing  60  in  the  column  of  days,  reason : 

"  In  60  da.  they  will  dig  a  ditch  60  times  as  long," 

and  write  60  in  the  numerator.  Placing  8  in  the  column  of  hours, 
reason  : 

"  Working  8  hr.  a  day,  the  length  will  be  8  times  as  great," 

and  write  8  in  the  numerator.  Placing  8  in  the  column  of  feet 
wide,  reason  : 

"  When  the  ditch  is  8  ft.  wide,  it  will  be  |  as  long," 

and  write  8  in  the  denominator.  Placing  3  in  the  column  of  feet 
deep,  reason  : 

"  When  the  ditch  is  3  ft.  deep,  it  will  be  1  as  long." 

and  write  3  in  the  denominator. 

The  operations  indicated  partially  at  (6)  and  completely  at  (c) 
should  all  be  written  at  (a),  the  other  two  being  introduced  merely  to 
show  how  the  work  proceeds. 

Find  the  result. 

16.  If  8  horses  consume  24  bushels  of  oats  in  16  days, 
how  long  will  352  bushels  last  11  horses  ? 

17.  If  a  piece  of  ground  160  rods  long  and  30^  rods  wide 
yields  15  bushels  of  wheat,  what  should  be  the  yield  of  a 
piece  121  rods  long  and  80  rods  wide  ? 

18.  If  12  men  in  9  days  build  a  wall  67^  feet  long  when 
the  wall  is  8  inches  thick,  how  many'  men  would  be  re- 
quired to  build  112i  feet  of  wall  in  5  days,  the  wall  to  be 
12  inches  thick  ? 

19.  If  256,000  bricks,  each  8  in.  x  4  in.  x  2  in.,  are  re- 
quired to  build  a  wall,  how  many  concrete  blocks,  each 
4  ft.  X  2  ft.  X  1  ft.,  would  be  required  for  the  same  wall, 
no  allowance  being  made  for  mortar  in  either  case  ? 

20.  If  17  men  in  12  days  earn  $612,  how  much  should 
be  the  earnings  of  11  men  in  14  days  ? 


Analysis  by   Aliquot  Parts  185 

21.  If  it  costs  $18  to  thresh,  the  wheat  raised  in  a  field 
60  rods  wide  and  80  rods  long,  what  should  it  cost  to  thresh 
the  wheat  raised  in  a  field  176  rods  long,  110  rods  wide  ? 

22.  A  farmer  paid  $140  for  material  required  to  fence 
a  field  80  rods  long  and  60  rods  wide.  What  would  be  the 
cost  of  the  material  required  to  inclose  a  field  176  rods  by 
110  rods  ? 

Note  the  difference  between  this  problem  and  the  preceding  one. 
The  yield  of  the  field  in  No.  21  depends  upon  an  area  60  rd.  by  80  rd.  ; 
the  cost  of  the  fence  in  No.  22  depends  upon  its  length,  60  rd.  +  80 
rd.  +  60  rd.  +  80  rd.  In  problem  21,  the  results  are  based  upon  the 
respective  products  of  60  x  80  and  110  x  176  ;  in  problem  22,  these 
numbers  are  addends,  and  not /actors. 

23.  If  it  costs  $180  to  cement  the  sides  and  the  bottom 
of  a  tank  24  ft.  long,  12  ft.  wide,  8  ft.  deep,  what  will  be 
the  cost  of  cementing  the  sides  and  the  bottom  of  a  tank 
48  ft.  by  24  ft.  by  16  ft.  ? 

24.  If  it  requires  12  hr.  20  min.  for  a  pipe  to  fill  a 
tank  24'  x  12'  x  8',  how  long  will  be  required  for  the  same 
pipe  to  fill  a  tank  48'  x  24'  x  16'? 

25.  If  8  horses  use  52i  bushels  of  oats  in  5  weeks,  how 
many  bushels  will  16  horses  use  during  April,  May,  and 
June? 

ANALYSIS   BY   ALIQUOT   PARTS. 

358.  Business  men,  as  a  rule,  dislike  long  operations  in 
multiplication  and  division.  Many  problems  usually  solved 
by  analysis  or  by  proportion,  they  work  by  aliquot  parts. 

359.  Written  Exercises. 

1.   If  24  men  receive  weekly  wages  of   $316.80,  what 
should  be  paid  30  men  ?  56  men  ? 

24  men  receive  ^316.80 
6  men  receive      78.20       i  of  24  men's  wages 
30  men  receive  §395.00     Ans. 


1 86  Arithmetic 

24  men  receive  $316.80 
24  men  receive    316.80 
8  men  receive    105.60      i  of  24  men's  wages 

2.  The  keeper  of  a  livery  stable  uses  on  an  average 
1080  lb.  hay  weekly  for  15  horses.  How  much  will  18  horses 
use  per  week  ?  12  horses  ?  20  horses  ? 

Add  I ;  subtract  | ;  add  \. 

3.  Find  the  cost  of  40  articles  at  $  29.70  per  gross  (144). 

144  cost  $29.70  or       144  cost  $29.70 

48  cost    $9.90     ^  gross  36  cost    $7,425     i  gross 

Deduct       8  cost      1.65    i  of  48    Add      4  cost         .825    ^  of  36 

4.  If  I  yd.  cloth  costs  $1.17,  what  is  the  cost  of  a  yard? 

3  fourths  costs  $1.17 

1  fourth   costs       .39     1  fourth  is  ^  of  3  fourths. 

5.  A  man  buys  a  f  interest  in  a  business  for  $2790. 
What  is  the  value  of  the  whole  business?  Of  ^  of  the 
business?     Off? 

I  is  worth  $2790        |  is  worth  $2790         (f)  j%  is  worth  $2790 
^  is  worth  I  is  worth  {^)  j\  is  worth 

1  is  worth  I  is  worth  (|)  ^^  is  worth 

6.  If  a  ^  interest  in  a  factory  is  worth  $8750  dollars, 
how  much  is  a  f  interest  worth?     A  j\  interest? 

7.  After  spending  \  of  his  money,  a  man  has  $84.69 
remaining.     What  had  he  at  first  ? 

8..  What  is  the  rent  of  a  house  for  3  yr.  5  mo.  18  da.  at 
$420  per  year? 

Rent  for  1  yr.  $420 
Rent  for  2  yr. 

Rent  for  4  mo.  i  of  $420 

Rent  for  1  mo.  :J  of  4  mo. 

Rent  for  15  da.  ^  of  1  mo. 

Rent  for  3  da.  ^  of  15  da. 

Rent  for  3  yr.  5  mo.  15  da.         $ 


Analysis  by  Aliquot  Parts  187 

9.    Find  the  rent  of  a  farm  for  2  yr.   7  mo.  20  da.  at 
$420  per  year. 

10.    A  man  paid  $1543.50  for  calico  at  the  rate  of  6^  per 

yard.     He  sold  it  for  7  ^  per  yard.     What  did  he  receive  for 

it? 

Cost  @  6^        $1543.50 


Profit  @\<P 


11.  Calico  which  cost  4i^  to  make  was  sold  by  the  manu- 
facturer at  6  ^  per  yard.  What  was  his  profit  on  a  sale  of 
%  1543.50  ?  What  was  the  cost  of  manufacturing  these 
goods  ? 

12.  A  quantity  of  goods  cost  $1543.50  at  6j^  per  pound; 
find  the  amount  received  for  them  at  5^  per  pound.  At 
5|^  ^  per  pound.     At  4^  per  pound. 

13.  A  lot  of  coffee  weighing  4864  lb.  was  sold  at  24^  per 
pound.     What  was  received  for  it? 

Cost  of  4864  lb.  @  25^  =  1  of  $4864  =  $1216. 
Deduct  cost  of  4864  lb.  @    1  j^  = 


Cost  of  4864  lb.  @  24^ 

14.  Find  the  cost  of  4864  lb.  coffee  : 

(a)   At  241^.  (c)  At24J^. 

(6)    At24j^.  {d)  At24ff^. 

15.  Find  the  cost  of  3248  yards  : 

(a)   At  121^.  id)  Atl3f^. 

(6)   At  131^.  (e)  At  12^. 

(c)   At  131^.  (/)  Atlli^. 

16.  Find  the  cost  of  8576  gallons  : 

(a)   At99L^.  (e)  At  87f/. 

(6)    At  491^.  (/)  At99J^. 

(c)    At  321^.  {g)  At88J^. 

id)   Atl9|^.  (/i)  At  861^. 


1 88  Arithmetic 


RATIO. 


360.  When  we  speak  of  the  ratio  of  two  numbers  or  quan- 
tities, we  mean  the  comparison  of  two  numbers  by  dividing 
the  first  by  the  second. 

361.  The  sign  of  ratio  is  the  colon  (:)  placed  between 
the  numbers  to  be  compared. 

To  express  the  ratio  of  4  to  12,  we  write  4 :  12.  This 
means  4  -j-  12,  or  ^^,  which  is  equal  to  \. 

In  Continental  Europe  the  colon  (:)  is  the  sign  both  of  ratio  and  of 
division,  our  sign  (-f-)  not  being  employed. 

362.  The  numbers  compared  constitute  a  couplet,  the  first 
term  of  which  is  called  the  antecedent,  and  the  second  the 
consequent. 

363.  Only  like  numbers  can  be  compared.  Thus,  the 
ratio  of  6  pecks  to  3  bushels  is  obtained  by  changing  the 
bushels  to  pecks,  so  that  the  couplet  becomes  6  pecks  to 
12  pecks. 

rr,,         , .     .     6  pecks        6       1 
TheratioiSj2^j^^  =  j2  =  2- 

364.  Fractions  may  be  compared  by  reducing  them  to  a  common 
denominator  and  comparing  their  numerators.  Thus,  the  ratio  of  |  to 
I  is  the  ratio  of  \^  to  |§,  or  15  to  16,  the  result  being  15  -^  16,  or  ^|. 

365.  Sight  Exercises. 
Find  the  ratios  of : 

1.  18  to  9. 

2.  12  bu.  to  4  bu. 

3.  $  4.50  to  %  15. 

4.  $14  to  $2. 

5.  15  mi.  to  105  mi. 


6. 

51  ft.  to  17  ft. 

7. 

45  to  90. 

8. 

28  in.  to  21  in. 

9. 

3  tenths  to  4  fifths. 

10. 

8  in.  to  2  ft. 

Ratio  189 

11.  What  is  the  ratio  of  2  quarts  to  a  bushel  ? 

12.  John  earns  f  1.50  per  day  and  Henry  earns  $2  per 
day.     What  is  the  ratio  of  their  respective  earnings  ? 

13.  Tea  costs  60^  per  pound  and  coffee  35^.  Find  the 
ratio  of  their  respective  prices. 

14.  A  earns  $9  in  the  same  time  that  B  earns  $12. 
What  is  the  ratio  of  A's  wages  to  B's  ? 

15.  A  man's  salary  is  $1500  a  year  and  his  expenses  are 
$900.     Find  the  ratio  of  his  expenses  to  his  salary. 

16.  A  boy's  age  is  15  and  his  father's  is  45. 

(a)  What  is  the  ratio  of  the  son's  age  to  that  of  his 
father  ? 

(b)  What  was  it  5  years  ago? 

(c)  What  was  the  ratio  ten  years  ago  ? 

(d)  What  will  it  be  in  5  years? 

(e)  What  will  the  ratio  be  in  15  years  ? 

17.  B's  age  is  to  A's  as  5  to  3.  How  old  is  A  if  B's  age 
is  20  years  ? 

18.  Give  3  couplets  whose  ratio  is  1  :  5. 

19.  Give  3  couplets  whose  ratio  is  5  to  2. 

INVERSE   RATIO. 

366  The  ratio  heretofore  considered  is  called  direct  ratio. 
The  ratio  of  18  men  to  6  men  is  3,  as  far  as  wages  are  con- 
cerned or  the  amount  of  work  done;  that  is,  18  men  will 
do  3  times  as  much  work  as  6  men,  or  18  men  will  earn  3 
times  as  much  money  as  6  men.  If,  however,  the  problem 
considers  the  time  taken  by  18  men  to  do  a  certain  work  as 
compared  with  the  time  taken  by  6  men,  the  ratio  is  ^. 
This  is  called  inverse  ratio. 


190  Arithmetic 

367.  Oral  Exercises. 

1.  What  is  the  ratio  of  6  horses  to  9  horses  as  to  the 
amount  of  work  done  ?  As  to  the  time  a  certain  quantity 
of  oats  will  last  ? 

2.  What  is  the  ratio  of  4j^  muslin  to  5  ^  muslin  as  to  the 
quantity  that  can  be  bought  for  a  certain  sum  ?  As  to  the 
cost  of  a  certain  number  of  yards  ? 

3.  One  wheel  is  9  feet  in  circumference  and  another  is 
12  feet.  Find  the  ratio  of  the  distances  covered  in  the 
same  number  of  revolutions  of  each ;  the  ratio  of  the 
number  of  revolutions  made  by  each  in  going  the  same 
distance. 

4.  Find  the  ratio  between  the  quantities  of  tea  and 
coffee,  respectively,  that  can  be  purchased  for  $4.20,  the 
former  being  worth  60  ^  per  pound  and  the  latter  35  ^. 

5.  What  is  the  ratio  of  the  quantity  of   60^  material 

and  the  quantity  of   35  ^  material  that  can  be  purchased 

for   a   dollar  ? 

PROPORTION 

368.  Two  couplets  having  equal  ratios  may  form  a 
proportion. 

369.  The  sign  of  proportion  is  the  double  colon  ( :  : ) 
placed  between  two  ratios.  The  sign  of  equality  (  =  )  is 
also  employed. 

370.  Since  4 : 8  is  equal  to  9 :  18,  the  ratio  in  each  case 
being  i  we  may  write  the  following  proportion : 

4  :  8  : :  9  :  18, 
which  is  read,  4  is  to  8  as  9  is  to  18. 

371.  The  foregoing  proportion  may  be  written  in  this 
form : 

4 :  8  =  9 :  18, 

4       9 
or  -  =  -^. 

8      18 


Proportion  191 

372.    The  four  numbers  (or  quantities)  constituting  a  pro- 
portion are  classified  as  : 

(a)  First  and  second  couplets. 

f  antecedents  the  first  and  third  terms. 

consequents  the  second  and  fourth  terms, 

f  extremes  the  first  and  fourth  terms. 

means  the  second  and  third  terms. 


(c)[ 


373.    If   three   terms    of    a    proportion    are    given,   the 
remainino^  term  can  be  ascertained 


'O 


374.  Sight  Exercises. 

Give  the  missing  number : 
.     3       ? 

5      15 

2.  J5  =  y  7.    5:15  =  15:?. 

7      21 

3.  ^  =  24*  ^'    '^••^•  =  21:24. 

.     ?      20 

4.  7.  =  :t7.-  9.    a?:9::20:36. 

,    1.5    8 

5.  Y^?'  •^^-    l-^:3::8:.T. 

375.  When  four  numbers  (or  quantities)  form  a  propor- 
tion, the  product  of  the  numbers  in  the  extremes  is  equal 
to  the  product  of  the  numbers  in  the  means. 

In  the  proportion 

3  yd.  :-4yd.  :  :  812:  816 
we  find  that 

3  X  16=4  X  12. 

376.  To  find  the  number  of  units  of  a  missing  extreme  in  a 
proportion,  divide  the  j)roduct  of  the  numbers  in  the  means  by 
the  number  of  units  in  the  given  extreme ;   and   to  find  the 


192  Arithmetic 

number  of  units  in  a  missing  mean,  divide  the  j^foduct  of  the 
numbers  in  the  extremes  by  the  number  of  units  in  the  given 
mean. 

377.  Written  Exercises. 
Find  the  missing  term  : 

1.  2  ft.  6  in.  :7  ft.  6  in.:  :25^:«^^. 

Reducing  the  terms  of  the  first  couplet  to  inches,  we  have 

30  in.  :90  in.:  :2b^:X^. 

™  ,  .     ^,         .     .       ^  .     90  X  25 

The  number,  x,  ni  the  missmg  term  is  : 

'    '  ^  30 

3 

90  X  25 
canceling,  we  have        x  =  — =  75.        Atis.  75^. 

2.  f6:fl8::a^T.:15T. 

3.  400  cu.  ft. :  1200  cii.  ft.  :  :  60  da. :  x  da. 

4.  9  men :  x  men  :  :  $  306  :  $  1020. 

5.  ?  days  :  14  days  :  :  108  rods  :  168  rods. 

6.  7  men  :  x  men  ::  22  A. :  66  A. 

7.  I  yd.  :1^  yd. :  :x:  9. 

8.  18  men  :  24  men  : :  108  da. :  x  da. 

9.  48  hr.  :45  1ir.  ::$aj:$15. 

10.    18  horses  :.T  horses::  142:  $14. 

378.  Oral  Problems. 

1.  If  I  pay  $  18  freight  for  the  transportation  of  15 
tons,  how  many  tons  should  be  carried  for  $6? 

2.  A  certain  number  of  men  can  build  400  cu.  ft.  of 
wall  in  60  days.  How  long  would  it  take  them  to  build 
1200  cu.  ft.? 

3.  If  2i  feet  of  wire  cost  25  cents,  what  should  be  paid 
for  7^  feet  at  the  same  rate  ? 

4.  If  9  yards  of  cloth  cost  $36,  how  many  yards  can 
be  bought  for  $52? 


Proportion  193 

5.  If  it  requires  14  days  for  a  certain  number  of  men 
to  dig  a  ditch  168  rods  long,  how  many  days  will  they 
require  to  complete  108  yards? 

6.  Seven  men  can  cut  22  acres  of  grain  in  a  certain 
time.  How  many  men  would  be  required  to  cut  66  acres  in 
the. same  time  ? 

7.  If  1^  yd.  ribbon  will  make  4  ties,  how  many  ties  will 
|-  yard  make  ? 

8.  Eighteen  men  receive  108  days  of  vacation  in  the 
summer.     How  many  days  will  24  men  receive  ? 

9.  A  man  is  paid  $  15  for  45  hours  of  work.  How  much 
should  he  be  paid  for  48  hours  of  work  ? 

10.    If  the  food  of  18  horses  costs  .^42  for  a  given  period, 
how  many  horses  can  be  fed  for  the  same  time  for  $  14  ? 

379.   Written  Problems. 

1.  If  |-  of  a  farm  is  worth  $5130,  what  is  the  value  of 
I  of  it  ? 

As  the  value  of  a  part  depends  directly  on  the  size  of  the  part,  the 
larger  the  part  the  greater  the  value  ;  the  following  is  the  proportion. 

Value  of  f  :  value  of  f  :  :  f  farm  :  |  farm, 
or  5130  :  a;  :  :  f  :  f. 

Therefore,  f  5C  =  5130  x  | 

or,  X  =  6130  X  I  -  f 

=  5130  X  f  X  |.     Cancel. 

2.  If  it  takes  810  yards  of  material  27  inches  wide  to 
make  a  certain  quantity  of  clothing,  how  many  yards  of 
material  30  inches  wide  would  be  required  to  make  the 
same  quantity  ? 

In  this  case  the  ratio  of  the  number  of  yards  required  is  in  inverse 

ratio  to  the  width,  the  greater  the  width  the  smaller  the  number  of 

yards  needed. 

Quantity  of  27  in.     Quantity  of  30  in. 

material  '        material 

9 

Canceling,  x  =  729.  Ans.  729  yd. 


1^4  Arithmetic 

Note.  For  convenience,  it  is  usual  to  make  the  missing  quantity 
the  fourth  term  of  the  proportion. 

3.  If  a  certain  quantity  of  food  lasts  65  men  12  days, 
liow  long  should  it  last  26  men  ? 

men         men  da.  da. 

26     :     65     :  :     12     :     X 

This  arrangement  shows  more  clearly  to  the  beginner  that  26  is  the 
divisor,  and  tliat  the  first  term  can  be  "canceled"  with  either  of  the 
other  two  given  terms. 

4.  The  number  of  boys  in  a  certain  school  is  to  the 
number  of  girls  as  8  is  to  9.  There  are  168  boys  in  the 
school.  How  many  of  the  pupils  are  girls  ?  How  many 
pupils  in  the  school  ? 

5.  Two  numbers  bear  to  each  other  the  ratio  of  5  to  12. 
The  larger  number  is  201 ;  find  the  other. 

6.  At  $  4.75  per  cord  of  128  cu.  ft.,  how  much  will  a 
pile  of  wood  be  worth  that  measures  16'  x  9'  x  1'  ? 

7.  If  a  man  travels  at  the  rate  of  10  miles  in  3  hours 
20  minutes,  how  far  can  he  travel  in  4  hours  40  minutes? 

8.  A  pole  6  feet  high  casts  a  shadow  of  4  ft.  8  in.  How 
high  is  a  tree  whose  shadow  at  the  same  time  measures  56 
feet  ? 

9.  If  42  men  require  18  days  to  do  a  certain  piece  of 
work,  how  many  men  will  be  required  to  do  the  same 
work  in  27  days  ? 

10.  If  49  acres  of  land  produce  2450  bu.  corn,  how  many 
bushels  should  be  produced  by  80  acres  ? 

11.  A  train  travels  224  miles  in  5  hours.  In  what  time 
can  it  make  a  trip  of  840  miles  at  the  same  rate  ? 

12.  In  raising  a  stone  by  means  of  a  crowbar,  a  pressure 
of  7  lb.  lifts  a  weight  of  49  lb.  How  much  pressure  is 
required  to  lift  a  weight  of  218  lb.  ? 


Proportion  195 

13.  If  9  barrels  of  lime  are  used  in  laying  8000  common 
bricks,  how  many  barrels  of  lime  will  be  required  in  build- 
ing a  wall  containing  120,000  bricks  ? 

14.  If  the  ratio  of  an  avoirdupois  pound  to  a  troy  pound 
is  7000  to  5760,  how  many  pounds  avoirdupois  are  equivalent 
to  175  pounds  troy? 

15.  If  the  ratio  of  an  avoirdupois  ounce  to  a  troy  ounce 
is  437|-  to  480,  how  many  ounces  avoirdupois  are  equal  to 
1000  troy  ounces  ? 

16.  A  cubic  foot  of  water  (1728  cu.  in.)  weighs  62J  lb. 
Find  the  weight  of  1  gallon  of  water,  231  cu.  in. 

17.  If  a  man  whose  property  is  valued  at  $  9000  pays 
$  82.80  taxes  per  year,  what  should  be  the  annual  taxes  of 
a  man  whose  property  is  valued  at  $  2250  ? 

18.  How  many  tons  of  2000  pounds  each  are  equal  to  50 
tons  of  2240  pounds  each  ? 

19.  If  the  profits  of  ,a  business  are  divided  between  two 
partners  in  the  ratio  of  12  to  15,  and  the  former  receives 
$  1440  of  the  profits,  how  much  should  the  latter  receive  ? 

20.  A  farmer  obtains  from  a  field  960  measured  bushels  of 
oats  weighing  28  pounds  to  the  bushel.  How  many  bushels 
can  he  sell  at  the  rate  of  32  pounds  by  weight  to  the 
bushel  ? 

PARTITIVE  PROPORTION. 

380.  The  process  of  dividing  a  given  number  into  parts 
proportional  to  given  numbers  is  called  partitive  proportion. 

381.  Written  Problems. 

1.  A,  B,  and  C  receive  $  360  for  hauling  a  quantity  of 
wood.  A  furnishes  2  teams,  B  3  teams,  and  C  4  teams. 
What  is  the  proportionate  share  of  each  ? 


196  Arithmetic 


2  teams 

3  teams 

4  teams 

9:2::$  360  :  A's  share. 
9  :  3  :  :  $  360  :  B's  share. 
9  :  4  :  :  $  360  :  C's  share. 


As  there  were  9  teams  at  work 
of  which  A  furnished  2,  the  whole 
sum  is  to  A's  share  as  9:2. 

The  whole  sum  is  to  B's  share  as 
9  :  3,  and  to  C's  share  as  9:4. 


2.  For  hauling  ice,  P,  Q,  and  E,  receive  $  210.  P  fur- 
nishes 2  teams  for  8  days,  Q  3  teams  for  10  days,  and  R  4 
teams  for  6  days.     How  should  the  money  be  divided  ? 


2x    8  =  16 

3  X  10  =  30 

4  X    6  =  24 

70  :  16  : 

:$210 

:  P's  share 

70  :  30  : 

:§210; 

;  Q's  share. 

70  :  24  : 

:$210: 

;  R's  share, 

P  furnishes  2  teams  for  8  days,  which  is  equivalent  to  1  team  for  16 
days.  Q  furnishes  3  teams  for  10  days,  which  is  equivalent  to  1  team 
for  30  days.  R  furnishes  4  teams  for  6  days,  which  is  equivalent  to  1 
team  for  24  days.  The  total  is  the  equivalent  of  70  days'  work,  of 
which  P,  Q,  and  R  are  entitled,  respectively,  to  ^§,  f§,  and  f^, 

3.  Divide  100  into  two  parts  which  shall  have  a  ratio  to 
each  other  of  6  to  9. 

4.  Divide  $  200  between  M  and  N  so  that  M  shall  re- 
ceive $  2  for  every  $  3  received  by  N. 

5.  X  and  Y  divide  between  them  $  300,  X  taking  -5^  of 
the  sum  taken  by  Y.     What  does  each  receive  ? 

6.  In  mixing  concrete  for  a  foundation  2  parts  of  cement, 
3  parts  of  broken  stone,  and  4  parts  of  sand  are  used.  How 
many  cubic  yards  of  each  are  required  to  make  2700  cubic 
yards  of  concrete  ? 

7.  Five  farmers  pay  $  75  for  the  expense  of  irrigating 
their  farms,  comprising  324  acres,  144  acres,  96  acres,  120 
acres,  and  216  acres,  respectively.  What  sum  should  be 
paid  by  each  ? 


Proportion  197 

8.  Six  families  agree  to  pay  a  teacher  on  the  basis  of  the 
respective  number  of  their  children  attending  the  school. 
How  much  should  each  contribute  to  the  $  200  paid,  if  the 
children  number  respectively  3,  1,  4,  2,  5,  and  1  ? 

9.  In  a  school  of  357  pupils  the  number  of  boys  is  to 
the  number  of  girls  as  8  to  9.  How  many  boys  in  the 
school  ?     How  many  girls  ? 

10.    Two  numbers  whose  sum  is  714  have  the  ratio  of 
5  to  12.     What  are  the  numbers? 


PARTNERSHIP. 

382.  Two  or  more  persons  desiring  to  combine  their 
capital  and  experience,  may  form  a  partnership  for  the  pur- 
pose of  carrying  on  business  of  any  kind. 

383.  Each  member  of  a  firm  taking  part  in  its  manage- 
ment is  called  a  general  partner.  In  some  cases,  one  or  more 
special  partners  are  admitted  to  a  firm. 

384.  The  sharing  of  the  profits  or  the  losses  among 
partners  is  a  matter  of  agreement.  In  the  problems  given 
under  this  heading,  it  is  assumed,  unless  otherwise  stated, 
that  profits  or  losses  are  shared  in  proportion  to  the  sums 
invested. 

385.  "Written  Exercises. 

1.  C,  D,  and  E  enter  into  partnership  contributing  re- 
spectively, $3500,  f  4500,  and  |5500.  The  profits  are 
$  7000.     What  share  of  the  profits  should  each  receive  ? 

2.  M  and  IST  form  a  partnership,  the  former  contributing 
$10,000  and  the  latter  $5000.  It  is  agreed  that  N  shall 
receive  $1000  from  the  profits  as  compensation  for  extra 
experience.  What  is  each  one's  share  of  the  profits  if  the 
year's  business  shows  a  profit  of  $5500  ? 


198  Arithmetic 

M  gets  his  share  of  $  4500. 

N  gets  his  share  of  $  4500,  and  $  1000  additional. 

3.  Three  men  investing  $4500,  f  5000,  and  $5500  re- 
spectively, sell  their  business  at  the  end  of  the  year  for 
$  11,250.     What  should  each  receive  as  his  share  ? 

4.  In  a  mining  enterprise  four  men  contribute  sums  of 
$2500,  $3000,  $3500,  and  $4000  respectively.  They  sell 
the  mine  for  $12,000  less  $300  commission.  What  is  each 
one's  share  of  the  proceeds? 

5.  M,  N,  and  0  raise  cattle  as  partners,  M  contributing 
$16,000,  N  $12,000,  and  O  $10,000.  Their  receipts  for 
the  year  are  $10,000  and  their  expenses  are  $2400.  What 
should  each  receive  from  the  profits  ? 

POWERS  AND  ROOTS. 

POWERS. 

386.  By  the  square  of  5  is  meant  the  employment  of  5 
twice  as  a  factor.  The  square  of  5  equals  5  x  5,  or  25.  It 
may  be  indicated  by  placing  a  small  2  above  the  5  to  the 
right,  thus,  51 

387.  The  cube  of  5  equals  5  x  5  x  5,  or  125.  It  is  indi- 
cated by  writing  a  small  3  above  the  5,  to  the  right,  which 
shows  that  5  is  to  be  taken  as  a  factor  3  times.  Thus,  5^  = 
125. 

388.  The  square  of  5  is  also  called  the  seco7id  power  of  5 
and  the  cube  of  5  is  called  the  tJiird  power  of  5. 

389.  The  small  number  employed  to  indicate  the  power 
is  called  an  exponent.  To  denote  that  5  is  to  be  raised  to 
the  fourth  power;  that  is,  that  5  is  to  be  taken  four  times  as 
a  factor,  we  use  the  exponent  4.  Thus,  5"*  =  5x5x5x5; 
5^  =  5x5x5x5x5. 


Powers  and   Roots  199 

390.  The  process  of  finding  the  power  of  a  given  number 
is  called  involution. 

391.  Oral  Exercises. 

Find: 

1.  The  square  of  6.  3.    The  fourth  power  of  3.* 

2.  The  cube  of  4.  4.    The  fifth  power  of  2. 

Give  the  values  of  the  following: 

5.  10-^  7.    6^  9.    10*  11.    5* 

6.  122  8.    20^  10.    12^  12.    3^ 

392.  Oral  Problems. 

1.  If  there  are  12  inches  in   a   linear   foot,  how   many 
square  inches  are  there  in  a  square  foot  ? 

2.  How  many  cubic  feet  in  a  cubic  yard? 

3.  How  many  square  feet  in  a  square  yard? 

4.  How  many  cubic  inches  in  a  cubic  foot  ? 

5.  How  many  square  yards  in  a  square  rod? 

393.  Find  the  square  of  20  +  5. 

The  product  of  (20  +  5)  by  (20  +  5)  is  equal  to  the  product  of  20 
times  (20  +  5)  added  to  the  product  of  5  times  (20  +  5). 

20  times  (20  +  5)  =     20-  +  20  times  5 
+  5  times  (20  +  5)  =  +  20  times  5  +  5- 

(20  +  5)  times  (20  +  5)  =     20--2  +  twice  (20  x  5)  +  S'-^ ; 
or  400  +  200  +  25,  or  625. 

The  square  of  the  sum  of  two  numbers  is  equal  to  the  square 
of  the  Jirst,  jjIus  tivice  the  product  of  the  first  by  the  second, 
plus  the  square  of  the  second. 

394.  Oral  Exercises. 

Give  answers: 

1.   202  +  2  X  (20  X  2)  -h  22  2.    (20  +  3)  x  (20  +  3) 


200  Arithmetic 

3.  312  5     352  7     3  52  9     ^^y 

4.  412  6.    452  8.    4.52  10.    (5L)2 

11.  A  square  field  measures  65  rods  on  a  side.  What  is 
its  area  in  square  rods? 

12.  How  many  square  feet  in  the  ceiling  of  a  square 
room,  one  side  of  which  is  42  feet  long? 

13.  The  floor  of  a  square  room  is  14  yards  wide.  How 
many  square  yards  does  it  contain? 

14.  What  is  the  area  of  a  floor  10^  yd.  long  and  10|^  yd. 
wide? 

15.  How  many  square  feet  in  the  six  equal  square  faces 
of  a  cube,  one  edge  of  which  measures  4  inches? 

ROOTS 

395.  To  find  a  root  of-  a  number  means  to  find  one  of  the 
equal  factors  of  the  number.  The  square  root  is  one  of  its 
two  equal  factors;  the  cube  root  is  one  of  its  three  equal 
factors. 

396.  The  square  root  of  64  is  indicated  thus,  V64.  The 
cube  root  is  indicated  thus,  V64.  The  4th  root  is  indicated 
thus,  V16. 

397.  Oral  Exercises, 

1.  Vioo  4.  Vl2i  7.  V25 

2.  V36  5.    V64        •  8.    V8i 

3.  V49  6.    V8i  9.    Vl44 

398.  Written  Exercises. 

1.    Find  the  square  root  of  1296. 
Resolving  1296  into  its  prime  factors,  we  have 

1296  =2x2x2x2x3x3x3x3. 


Powers  and   Roots  201 

Separating  these  factors  into  two  sets  containing  the  same  factors, 
we  have  1296  =  (2  x  2  x  3  x  8)  x  (2  x  2  x  3  x  3) 

Or,  1296  =  36  X  36,  or  (36)2 

Therefore,     \/l296  -  36.        Ans. 

By  the  use  of  factors,  find  the  square  root  of  each  of  the 
following: 

2.  196  4.    256  6.    225 

3.  324  5.   441  7.    484 

8.   Find  the  cube  root  of  1728. 

1728  =2x2x2x2x2x2x3x3x3 
Separating  these  factors  into  three  sets  containing  the  same  factors, 
we  have  1728  =  (2  x  2  x  3)  x  (2  x  2  x  3)  x  (2  x-  2  x  3) 

1728  =  12  X  12  X  12,  or  (12)3 

Therefore,      v'T728  =  12.         Ans. 

By  the  use  of  factors  find  the  cube  root  of  each  of  the 
following: 

9.   216  11.    729  13.   3375 

10.   576  12.    2744  14.   4096 

Extract  the  roots  indicated : 


15.    V1296  16.    \/1024  17.    V729 

SQUARE   ROOT. 
399.    Written  Exercises. 

1.    Extract  the  square  root  of  1225. 

Since  the  square  of  30  Ls  900,  and  the  square  of  40  is  1600,  the 
square  root  of  1225  is  between  30  and  40  ;  that  is,  it  equals  30  + 
another  number.     Calling  the  missing  number  3",  we  have 
(30  +  N)  X  (30  +  .V)  zz  1225, 
or  30-'  +  (2  X  30  x  .Y)  +  .V^  =  i225 

Deducting  30'^,  or  900,  from  each  side  of  the  equation,  we  have 

(60  X  .V)  +  A^2  ^  1225  -  900  =  325 
This  may  be  changed  to  read  as  follows: 

60  times  N  +  X  times  N  =  325, 
or  (60  +  X)  times  N=  325. 


202  Arithmetic 

The  problem  now  becomes 

"  What  number  added  to  60  and  the  sum  multiplied  by  the  same 
number  equals  325?"     Using  60  as  a  trial  divisor,  we  find   that  5 
satisfies  the  conditions  ;  viz.,  (60  +  5)  x  5  =  326. 
The  process  may  be  shown  in  this  manner 

1225(30  +  5 
900 
Trial  divisor  60      326 
(60  +  5)  X  6       325 


2  4    9     Ans.   249 


6'20'01 

4 
44  220 

176 

489  4401 

4401 

2.    Find  the  square  root  of  62,001. 

In  practice,  the  work  is  shortened  by  the 
omission  of  ciphers.  The  number  is  pointed 
off  in  periods  of  two  places  each,  beginning 
at  the  right. 


The  square  root  of  the  greatest  square  in  the  first  period  is  2,  which  is 
written  above  the  6,  and  its  square,  4,  is  written  under  the  6.  Taking 
4  from  6  gives  2  as  a  remainder,  after  which  are  placed  the  two  figures 
of  the  second  period,  making -the  next  dividend  220.  The  2  in  the  root  is 
doubled,  giving  4  as  a  trial  divisor,  to  which  is  to  be  annexed  the  second 
figure  of  the  root.  This  is  found  to  be  4,  which  is  written  above  the 
next  period,  and  also  after  the  4  of  the  trial  divisor,  which  now  be- 
comes 44. .  Multiplying  the  latter  by  the  4  in  the  root  gives  a  product 
of  176,  which  is  deducted  from  220  leaving  a  remainder  of  44.  To  this 
are  annexed  the  two  figures  of  the  last  period,  making  the  next  divi- 
dend 4401.  The  24  in  the  result  is  doubled,  giving  48  as  a  trial  divisor, 
to  which  is  to  be  annexed  the  third  figure  of  the  root.  This  is  found 
to  be  9,  which  is  written  above  the  third  period  and  also  after  the  48 
of  the  trial  divisor,  which  now  becomes  489.  Multiplying  the  latter 
by  the  9  in  the  root,  gives  a  product  of  4401.  There  being  no  re- 
mainder, 249  is  the  square  root  of  62,001. 


3.    Find  the  square  root  of  164,025. 


As  the  first  trial  divisor,  8  with  a  figure 

annexed,  is  not  contained  in  40,  another 

period  must  be  brought  over,  the  new  trial 

divisor  being  80.  ^^^      ^^  ^''^ 

40  25 


4   0    6         Ans.  405 


16'40'25 
16 


Powers  and   Roots  203 

Find  the  square  root  of  the  following : 

4.  169  6.   529  8.   123.21 

5.  189  7.    616  9.   492.84 

400.  In  pointing  off  numbers  containing  decimals,  com- 
mence at  the  decimal  point  and  point  off  two  figures  each 
way,  annexing  decimal  ciphers  when  necessary. 

401.  Find  the  square  root  of  the  following  (3  figures  in 
the  result) : 

1.  2  3.      .3  (.30)  5.    10. 

2.  3  4.      .5  9.    12. 

402.  Written  Problems. 

1.  How  many  rods  long  is  a  square  field  containing  10 
acres  ? 

2.  How  many  yards  wide  is  a  square  plot  of  ground  con- 
taining 1  acre  ? 

3.  A  cube  has  6  square  faces,  and  its  entire  surface  con- 
tains 1350  square  inches.     What  is  the  volume  of  the  cube? 

4.  What  is  the  side  of  a  square  field  that  will  contain 
the  same  area  as  a  rectangular  field  81  rods  wide  by  144 
rods  long  ? 

5.  A  man  has  exchanged  two  plots,  one  240  rods  square 
and  the  other  320  rods  square,  for  a  single  square  plot  equal 
in  area  to  the  combined  areas  of  his  two  plots.  What  is 
the  length  of  a  side  of  the  new  plot  ? 

APPLICATIONS   OF   POWERS   AND   ROOTS. 

403.  Draw  a  right-angled  triangle,  ABC,  having  the  base 
BC,  the  perpendicular  AB,  and  the  hypotenuse  AC.  On 
the  hypotenuse  AC,  construct  a  square,  and  extend  BA  and 
BC  to  form  two  sides  of  a  square  circumscribing  the  one 
constructed  on  AC.     Complete  this  square,  then  extend  the 


204 


Arithmetic 


upper  and  lower  sides  to  the  right  and  construct  a  second 
square  equal  in  area  to  the  other.     The  iirst  large  square 

contains  the  square  M  and 
4  equal  right-angled  tri- 
angles marked  X.  The 
other  large  square  contains 
the  squares  N  and  0  and  4 
equal  right-angled  triangles 
marked  X.      Taking  away 


\y 

Ax 

M 

A 

/ 

0 

'i^ 

A 

^   / 

\y 

/. 

N 

/ 
/ 

/    ^ 

B 


the  four  right-angled  triangles  from  each,  we  have 
square  M=  square  iV+  square  0. 
M  is  the  square  constructed  on  the  hypotenuse  of  the 
right-angled  triangle,  N  is  the  square  constructed  on  the 
perpendicular  of  the  right-angled  triangle,  and  0  is  a  square 
constructed  on  the  base  of  a  right-angled  triangle,  of  the 
same  dimensions  as  the  triangle  ABC. 

The  square  constructed  on  the  hypoteymse  of  a  right-angled 
triangle  is  equal  to  the  sum  of  the  squares  constructed  on  the 
other  two  sides. 

404.  Calling  the  unit  of  measurement  of  the  hypotenuse 

H,  and  the  units  of  the  other  sides  0  and  B  respectively, 

we  have 

H''  =  P'-\-B\ 

or  H=  ^P'  +  B\ 

That  is,  the  length  of  the  hypotenuse  is  equal  to  the 
square  root  of  the  sum  of  the  squares  of  the  other  two 
sides. 

To  find  the  length  of  the  hypotenuse  of  a  right-angled  tri- 
angle, add  together  the  squares  of  the  lengths  of  each  of  the 
other  sides  and  extract  the  square  i^oot  of  the  sum. 

405.  Since  P'-{-B'  =  H% 

B^  =  H'-F', 

and  P'  =  H'-B\         

Therefore,  B  =  ^ H'  -  P'  and  P=VJP-  BK 


Powers  and   Roots  205 

To  find  the  length  of  the  base  (or  the  perpendicular)  of  a 
right-angled  triangle,  extract  the  square  root  of  the  difference 
betweeri  the  sqiiare  of  the  length  of  the  hypotenuse  and  the 
square  of  the  length  of  the  other  side. 

406.  Written  Exercises. 

Find  the  length  of  the  missing  side: 

1.  P=^  rods,  B  =  12  rods. 

2.  H=1S  yards,  P  =  5  yards. 

3.  P  =  24  feet,  B  =  7  feet. 

4.  5  =  16  miles,  P=  30  miles. 

5.  11  =  4:5  inches,  B  =  36  inches. 

6.  H=4\  inches,  P=3^  inches. 

7.  5  =  40  feet,  P=d  feet. 

8.  H=61  yards,  JB  =  60  yards. 

9.  5  =  40  rods,  P=  40  rods. 
10.  Zr=  100  feet,  B  =  P. 

407.  "Written  Problems. 

Note.     Make  diagrams  when  necessary. 

1.  A  man  travels  due  north  48  miles,  then  due  east 
64  miles.  How  far  is  he  in  a  straight  line  from  his  starting 
point  ? 

2.  Find  the  distance  between  the  opposite  corners  of  the 
floor  of  a  room  24  feet  long  18  feet  wide. 

3.  A  rectangular  field  is  32  rods  long  and  60  rods  wide. 
How  many  acres  does  it  contain  ?  What  is  the  length  of 
the  diagonal  ? 

4.  A  triangle  whose  base  is  60  feet  has  two  equal  sides 
measuring  34  feet  each.  What  is  the  length  of  the  perpen- 
dicular let  fall  from  the  apex  to  the  middle  of  the  base  ? 


2o6  Arithmetic 

5.  A  square  field  contains  40  acres.  What  is  the  length 
of  each  side  ?  Find  the  square  of  the  number  that  repre- 
sents the  length  of  the  diagonal  in  rods. 

6.  If  the  square  of  the  number  of  rods  in  the  diagonal 
of  a  square  field  is  1600,  what  is  the  square  of  the  number 
of  rods  in  each  side  ?     How  many  acres  in  the  field  ? 

7.  The  diagonal  of  a  square  field  is  25  rods.  Find  the 
area  of  the  field  in  acres  ?  ^ 

10  sq.  chains  =  1  acre. 

8.  A  room  is  12  feet  long,  9  feet  wide,  and  8  feet  high. 
How  long  is  the  diagonal  of  the  floor?  How  far  is  the 
end  of  this  diagonal  from  the  opposite  corner  of  the  ceiling  ? 

9.  How  far  from  the  foot  of  a  house  must  a  51-foot 
ladder  be  placed  so  that  it  will  just  reach  a  window  45  feet 
above  the  street? 

10.  A  ladder  50  feet  long  is  so  placed  between  two  houses 
that  it  can  just  reach  the  top  of  each  without  being  moved 
at  the  foot.  One  house  is  48  feet  high  and  the  other  is  40. 
How  far  apart  are  the  houses  ? 


CHAPTER   VII. 

MENSURATION;   MISCELLANEOUS   PROBLEMS. 
AREAS   OF  PLANE   SURFACES. 

408.  To  find  the  area  of  a  plane  surface  is  to  ascertain 
how  many  times  this  surface  contains  another  surface  taken 
as  a  unit,  or  how  many  aliquot  parts  of  this  other  surface  it 
contains. 

409.  As  a  rule,  the  unit  employed  as  the  measure  of  a 
surface  is  a  square  having  for  its  sides  the  unit  of  length. 
Thus,  when  the  unit  of  length  is  the  inch,  the  unit  of  sur- 
face is  the  square  inch;  when  the  unit  of  length  is  the  foot, 
the  unit  of  surface  is  the  square  foot ;  etc. 

The  acre  is  the  only  unit  of  surface  that  has  not  a  corresponding 
unit  of  length. 

410.  A  plane  figure  bounded  by  straight  lines  is  called  a 
polygon.  A  polygon  of  three  sides  is  called  a  triangle;  of 
four  sides,  a  quadrilateral ;  of  five  sides,  a  pentagon ;  of  six 
sides,  a  hexagon. 

A  triangle  containing  a  right  angle  is  called  a  right-angled  triangle^ 
or  a  right  triangle. 

411.  Considering  the  lengths  of  the  sides,  a  triangle  hav- 
ing three  equal  sides  is  called  an  equilateral  triangle;  one 
having  only  two  equal  sides  is  called  an  isosceles  triangle ; 
one  in  which  all  the  sides  are  unequal,  is  called  a  scalene 
triangle. 

207 


io8 


Arithmetic 


412.  A  parallelogram  is  a  quadrilateral  whose  opposite 
sides  are  parallel  (Figs.  1-4). 

A  parallelogram  having  four  square  corners  is  called  a 
rectangle  (Figs.  1  and  2).     When  the  four  sides  of  a  rec- 


FlG.  1. 


Fig.  2. 


Fig.  3. 


Fig.  4. 


tangle  are  equal,  it  is  called  a  square  (Fig.  1) ;  when  the 
adjacent  sides  are  luiequal,  the  term  oblong  is  frequently 
applied  (Fig.  2). 

413.  The  rhombus  (Fig.  3)  and  the  rhomboid  (Fig.  4)  have 
no  square  corners ;  in  the  former  the  sides  are  all  equal;  in 
the  latter  the  adjacent  sides  are  unequal. 

414.  A  quadrilateral  having  only  two  of  its  sides  parallel 


Fig.  5. 


Fig.  6. 


is  called  a  trapezoid  (Fig.  5) ;  one  having  no  parallel  sides  is 
called  a  trapezium  (Fig.  6). 

Preliminary  Exercises. 

415.  1.  Cut  from  a  strip  of  paper  2  inches  wide  a  rhom- 
bus ABDC  having  its  sides  3  inches  long.  The  altitude 
is  AX. 

A  perpendicular  that  measures  the  distance  between  two  parallel 
sides  of  a  quadrilateral  is  called  the  altitude. 

2.  Cut  from  a  strip  of  paper  2  inches  wide  two  rhom- 
boids having  two  parallel  sides  each  3  inches  long,  the  remain- 
ing two  sides  of  one  rhomboid  measuring  2i  inches  each, 
and  those  of  the  other  rhomboid  measuring  3J  inches 
each.     What  is  the  altitude  of  each  rhomboid  ? 


Mensuration 


209 


B 


3.  From  one  corner  of  each  of  the 
three  parallelograms  draw  a  perpen- 
dicular to  the  opposite  side,  and  cut 
off  a  right  triangle  {ACX).  Transfer 
the  triangle  to  the  other  side  (BDY). 
What  are  the  dimensions  in  each 
case  of  the  rectangle  thus  formed  ? 


The  number  of  square  units  in  the  area  of  a  parallelogram 
is  equal  to  the  product  of  the  number  of  units  in  the  base  by 
the  number  in  the  altitude. 

Note.  The  dimensions  must  be  expressed  in  the  same  units  before 
performing  the  multiplication.  • 

4.  From  a  paper  rectangle  3  inches  by  2  inches,  ABDC, 
cut  a  triangle  YCD,  Y  being  taken  at  any 
point  on  AB.  Place  the  remaining  tri- 
angles on  the  triangle  YCD  to  show  that 
together  they  are  equal  to  YCD  and  that 
the  triangle  YCD,  therefore,  is  one-half 
of    the   rectangle   ABDC.      How   many 

square  inches  are  there  in  the  area  of  the  rectangle  ?    AYhat 

is  the  area  of  YCD  ? 

Tlie  number  of  squai'e  units  in  the  area  of  a  triangle  is 
equal  to  one-half  the  product  of  the  number  of  units  in  the 
base  by  the  number  in  the  altitude. 


The  line  that  measures  the  altitude  of 
a  triangle  may  lie  outside  of  the  triangle. 

In  the  accompanying  figure,  consider- 
ing BC  as  the  base  of  the  triangle  ABC^ 
the  altitude  is  AX.  Its  length,  however, 
is  the  same  as  that  of  MB  or  XC. 

The  triangle  ABC  =  h  parallelogram  ^F^C 

Area  =  *  (^C  x  AX) 


^  rectangle  iO/5C. 


2IO 


Arithmetic 


Any  side  of  a  triangle  or  of  a  parallelogram  may  be  taken 
as  the  base,  the  altitude  in  each  case  being  the  perpendic- 
ular let  fall  on  the  base  (or  the  base  produced)  from  the 
vertex  of  the  opposite  angle. 

Note.  The  dimensions  of  a  triangle  or  of  a  parallelogram  are  the 
base  and  the  altitude,  as  these  determine  the  area,  whatever  may  be 
the  shape  of  the  figure. 

416.  Cut  from  a  strip  of  paper  2  inches  wide  several 
trapezoids,  making  one  parallel  side  2i  inches  long  in  each 

case  and  the  other  3^ 
inches  long. 

Fold  AB  over  on  DC, 
creasing  the  paper  along 
XY.  Measure  XF.  Cut 
off  the  triangle  Y3C 
and  place  it   at  5  2  Y. 


1 

B 

2 

\  6  1 

\ 

a 
\ 

\! 

\ 

i<. 

■^\  ~ 

\  \ 

1        N 

1     a 

D 


Cut  off  ^  1 X  and  place  at  X  4  Z>. 


What  are  the  dimensions  of  the  rectangle  thus  formed  ? 


A  trapezoid  is  equal  in   area  to   a   rectangle   whose  di- 
mensions are  the  altitude  of   the 

trapezoid  and  the  half  sum  of  the  ^  ^^          ^ 

parallel  sides. 

The  trapezoid  RSTU  is  divided 
into  two  triangles  by  RT. 


N  T 


Area  of     BTS  =  ^  RS  X  MT 
Area  of     RUT  =  ^  UT  x  MT(RN) 
Adding,  Area  of  RSTU  =  i  {RS  +  UT)  x  MT 


The  number  of  square  units  in  the  area  of  a  trapezoid  is 
equal  to  the  product  of  one  half  the  sum  of  the  units  in  the 
parallel  sides  by  the  number  in  the  altitude. 


Mensuration  211 


417.  In  the  trapezium  FOHI,  a 
diagonal  GI  is  drawn,  and  perpen- 
diculars Fa  and  Hh  are  let  fall 
upon  it  from  the  vertex  of  each  of 
the  opposite  angles.  The  trapezoid 
is  thus  divided  into  two  triangles 
whose  areas  are  as  follows : 


Area  of     FGI  =  IG  x  i  Fa 
Area  of     GHI  =  IG  x  \  Hb 


Area  of  FGHI  =  IG  x  i  (Fa  +  Hb) 

The  number  of  square  units  in  the  area  of  a  trapezoid  is 
equal  to  the  product  of  the  number  of  units  in  the  diagonal 
by  one  half  the  sum  of  the  units  in  the  two  altitudes  let  fall  on 
the  diagonal. 

AREA   OF   A   POLYGON. 

418.  Written  Problems. 

Note.  In  some  of  the  following  problems  the  length  of  a  side  is  to 
be  obtained  from  a  given  area  or  other  data.  In  others,  a  missing 
dimension  must  be  calculated  before  a  required  area  can  be  obtained. 
The  employment  of  a  diagram  on  which  are  noted  the  items  given  will 
frequently  aid  the  pupil  to  determine  the  steps  necessary  to  be  taken. 
The  only  difficulty  in  these  problems  consists  in  determining  the 
operations  required. 

1.  Two  rectangular  fields  are  equal  in  area;  one  measures 
73.10  rods  by  28.80  rods;  the  length  of  the  second  is  59.80 
rods.     What  is  the  width  of  the  second  field  ?     (Cancel.) 

2.  How  many  acres  in  a  rectangular  field  having  one 
side  75  rods  long,  and  its  diagonal  measuring  125  rods  ? 

3.  A  classroom  30  feet  long  has  a  floor  space  of  600 
square  feet.  It  is  desired  to  increase  the  width  of  the  room 
to  accommodate  40  boys,  giving  each  2  square  yards  of  floor 


212 


Arithmetic 


space.     How  many  feet  must  be  added  to  the  width  of  the 
room  ? 

4.    A  12-acre  field  has  the  form  of  an  isos- 
celes triangle  whose  altitude  measures  60  rods. 
Find  the  sides  of  the  triangle. 
60BC 


2 


1920. 


BC=?     CB  =?    AC  =  ^6{f  +  CB^ 


5.  How  many  yards  of  fencing  will  be 
required  to  inclose  a  plot  of  ground  in  the 
form  of  a  right  triangle  having  a  perpendicular 
of  12  rods  and  containing  ^  acre  ? 


X  rd. 


2  X  Base 


D 


B 


=  96.     Find  hypotenuse. 

6.  How  many  square  yards  are  there  in  the 
gable  end  of  a  house,  the  measurements  being : 
AB,  30  feet;  AE,  36  feet;  DE  and  DC,  each  25 

feet? 

7.  A   pentagonal   field   has  a   base   AE,  100 
rods     long.      Perpendicu- 
lars    to    AE     from     the 

other  three  corners  measure  as  fol- 
lows:  Bl,  60  rods;  02,  80  rods; 
DS,  70  rods.  The  points  1,  2,  and  3 
are,  respectively,  15,  50,  and  80  rods 
distant  from  A.  How  many  square 
rods  does  the  field  contain  ? 

8.  Owing  to  the  swampy  ground  it  is  difficult  to  ascer- 
tain the  length  of  MQ  by  actual 
measurement.  The  other  sides  have 
lengths  as  follows :  JOT,  17  rods ; 
NO,  5  rods;  OP,  10  rods;  PQ,  13 
rods.  The  following  are  the  lengths 
Q    of  the  perpendiculars  to  MQi   Na^ 


M^-B^ 


Mensuration  213 

15  rods ;    Oh,  18  rods  ;  Pc,  12  rods.     Find  the  area  of  the 
field,  and  the  length  of  MQ. 

Ma  is  the  base  of  a  right-angled  triangle,  the  perpendicular  being 
Ma,  and  MX  the  hypotenuse.  Using  OX  and  01,  the  length  of 
JVl  (or  its  equal  ah)  can  be  ascertained ;    etc. 

9.    Find  the  altitude  of  an  equilateral  triangle  having 
sides  of  6  feet  each.     Find  its  area. 

10.  Find  the  altitude  and  the  area  of  an  equilateral 
triangle  each  of  whose  sides  measures  1  foot. 

To  find  the  Area  of  a  Triangle  ivhen  the  Lengths  of  the  Sides 

are  Given. 

419.  In  many  cases  a  person  desirous  of  ascertaining  the 
contents  of  a  triangular  piece  of  ground  finds  it  difficult  to 
locate  and  measure  the  altitude.  In  such  a  case  the  area  is 
obtained  from  the  lengths  of  the  sides  by  the  following 
formula :  Calling  half  of  the  perimeter  h,  and  the  sides  a, 
hj  and  c,  respectively, 


Area  =  V/t  X  {Ji  —  a)  x  (h  —  b)x(h  —  c). 

420.  TJie  number  of  square  units  in  the  area  of  a  triangle 
is  equal  to  the  number  of  units  in  the  square  root  of  the  com- 
bined product  of  the  units  in  one  half  of  the  perimeter  by  the 
respective  differences  between  the  units  in  the  half  perimeter 
and  the  units  in  each  side,  successively. 

421.  The  following  examples  will  illustrate  the  method: 
1.   Find  the  area  of  a  triangle  whose  sides  measure  16, 

20,  and  28  yards,  respectively. 

71  =  1(16  +  20  +  28)  =S'2 
h-a  =  32-16  =  16 
h-b  =  S2-'20  =  l2 
h-c  =  S2-2S=    4: 


Area  =  V32  x  16  x  12  x  4  zr  \/24756  =  156.76 
Ans.  156.76  sq.  yd. 


214  Arithmetic 

2.  The  sides  of  a  triangular  field  measure  100  rods,  156 
rods,  and  224  rods,  respectively.  How  many  acres  does  the 
field  contain  ? 

i  of  (100  +  166  +  224)  =  240 
240  -  100  =  140 
240  -  156  =  84 
240  -  224  =    16 

Area  in  square  rods  =  \/240  x  140  x  84  x  16  =  6720 

Area,  6720  sq.  rd.  =  42  acres.  Ans.  42  acres. 

3.  The  boundaries  of  a  triangular  farm  measure  340 
rods,  200  rods,  and  420  rods,  respectively.  How  many  acres 
are  there  in  the  farm  ? 

4.  Find  the  number  of  square  rods  in  a  triangular  piece 
of  woodland  whose  sides  are,  respectively,  52  rods,  56  rods, 
and  60  rods. 

5.  How  many  acres  are  contained  in  a  triangular  park 
whose  sides  are  52  rods,  148  rods,  and  160  rods,  respectively? 

6.  A  piece  of  ground  in  the  form  of  a  triangle  has  sides 
of  45  feet,  111  feet,  and  132  feet,  respectively.  How  many 
square  yards  does  it  contain  ? 

7.  Find    the  .  area    in    square   rods   of    a   quadrilateral 

measuring  as  follows : 

AB,  14  rods  ;  BC,  20  rods  ;  CD, 
21  rods;  and  DA,  15  rods.  The 
length  of  the  diagonal  BD  is  13 
rods. 

Note.  Find  the  area  of  each  triangle  separately.  Make  a  dia- 
gram and  wi'ite  on  each  side  its  length. 

AREA  OF  A  REGULAR  HEXAGON. 
422.   Exercises. 

1.  Find  the  altitude  of  an  equilateral  triangle  whose 
sides  are  1  inch.     (See  cut  on  next  page.) 


Mensuration 


215 


Ax=  Vab''  -  Bx^  =  vr^  =  V|  = 
V3  ^  V3  ^  1  r, 

2.  What  is  the  area  of  an  equilat- 
eral triangle  whose  sides  are  each  4 
inches  ? 


The  area  may  be  obtained  by  finding  the  altitude  and  multiplying 
this  by  one  half  the  base,  or  by  the  method  given  in  Art.  420. 


Area  in  square  inches  =  \/6  x  2  x  2  x  2. 

3.  Find  the  area  of  a  hexagon  composed  of  six  equal 
equilateral  triangles,  each  side  of  a  triangle  measuring  2 
inches. 

The  apothem  of  a  regular  polygon  is  the  line  drawn  from  the  center 
of  the  polygon  to  the  middle  point  of  one  side. 

4.  Find  the  apothem  of  a  regular  hexagon  whose  side  is 

1  inch.     (See  No.  1.) 

5.  Find  the  apothem  of  a  regular  hexagon  whose  side  is 

2  inches.     AVhose  side  is  4  inches. 

The  area  of  a  regular  polygon  =  1  (perimeter  x  apothem) . 

6.  How  many  square  feet  of  surface  will  be  covered  by 
1000  hexagonal  tiles,  each  6  inches  on  a  side  ? 

UNITED    STATES   PUBLIC    LANDS. 

423.  Land  owned  by  the  United  States  government  is 
laid  out  in  townships  six  miles  square,  which  are  divided 
into  sections  one  mile  square,  containing  G40  acres. 

424.  Townships  are  located  with  reference  to  a  line  called 
the  principal  meridian  and  a  base  line  intersecting  the 
former  at  right  angles.  The  townships  are  bounded  by 
lines  running  north  and  south,  6  miles  apart,  parallel  to  the 
principal  meridian,  and  by  east  and  west  lines,  also  six 
miles  apart,  parallel  to  the  base  line. 


2l6 


Arithmetic 


425.    The  rows  of  townships  bounded  on  one  side  by  the 
principal  meridian  are  called  Range  1,  East,  and  Range  1, 

West,  according  to  their 
location,  the  next  rows 
being  Range  2,  East,  and 
Range  2,  West,  etc.  The 
rows  on  either  side  of 
the  base  line  are  desig- 
nated Township  1,  North, 
and  Township  1,  South. 
The  Township  marked 
(a),  Fig.  1,  is  described  as 
Township  3  South,  Range 
3  East,  abbreviated  to  T. 
3  S.,  R.  3  E.  Township 
(6)  is  described  as  T.  4  N., 
R.  2  W. 


1 1 

TOWNSHIP 

5 

1 
NORTH 

TOWNSHIP 

4 

■D 

NORTH 

f, 

3D 

■33 

3J 

3) 

:n 

x 

Z 

33 

3J 

33 

33 

->— 

^>— 

->- 

->— 

->— 

*'•':;; 

->- 

->— 

— >- 

->- 

7 

■jL 

Z. 

7 

7 

Z 

7 

Z 

7 

o 

o 

o 

o 

O 

Uz 

o 

o 

C) 

CD 

-m- 

-m- 

-m- 

-m- 

-rn- 

->i^ 

km- 

-m- 

— m- 

-n- 

CJl 

-t>- 

co 

ro 

-* 

r- 

N3 

co 

■<^ 

CJi 

BA 

SE 

> 

CD 

LIT 

ME 

m 

m 
-co— 

H 

m 
-co— 

H 

m 
-co- 

H 

m 

CO 

H 

.m- 

-D 

m 

m 
> 

-CO- 

H 
a 

m 
> 

_co_ 

H 

> 
_co_ 

H 

TO 

WNP 

HIP 

4 

> 

SOL 

JTH 

_^ 

1 

TOWNSHIP 

1         1 

5 

SOUTH 

Fig.  1. 


TOWNSHIP 

N 


426.    The    townships    are    subdivided    into    36    sections 

by    lines    a    mile    apart,    running 

due  north   and    south,  which  are 

crossed  by  lines  running  east  and 

west.       They    are    numbered    as 

shown  in 
Fig.  2,  ^ 
No.  1  be- 
ing always 
the  north- 
east sec- 
tion. 


SECTION 

N 
ONE  MILE 


W 


N 

w 

V4 

N.  E.  ^ 

UJ 

160  A. 

E 

UJ 

z 
o 

S. 

w. 

V4 

W.i^ 
of 

S.E.i^ 
80  A. 

:v.E.j< 

ofS.K.i^ 
40  A. 

ofS.K.Ji 
4U  A. 

r 

5 

4 

3 

2 

1 

7 

8 

9 

10 

n 

12 

18 

17 

16 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

^  E 
>< 


SIX  MILES 
S 

Fig.  2. 


ONE  MILE 
S 

Fig.  3. 


427.    The    section    contains    640 
acres.     When  subdivided,  the  parts 


are  described  in  the  manner  shown 
in  Eig.  3.      A    half   section    of   320 
acres  may  be  a  North  half,  a  South  half,  an  East  half,  or  a, 


Mensuration 


217 


West  half,  according  to  its  location.  The  ISTorth  half  in 
Fig.  3  is  divided  into  two  quarter  sections,  as  is  the  West 
half. 

428.  Sight  Exercises. 

1.  What  is  the  value  of  S.E.  i  of  Sec.  27,  T.  7  N.,  R.  15 
W.,  at  $20  per  acre? 

2.  How  many  square  rods  in  S.E.  i  of  S.W.  ^  of  Sec. 
25,  T.  5  S.,  R.  15  W.  ? 

3.  How  many  rods  of  fence  will  be  required  to  inclose 
S.W.  1  of  S.W.  i  of  the  same  section  ? 

4.  A  farmer  plants  the  W.  ^  of  S.W.  i  of  the  same  sec- 
tion in  corn.  How  many  bushels  of  corn  does  he  obtain  to 
the  acre,  if  the  total  yield  is  4000  bushels  ? 

5.  If  the  owner  sold  the  West  half  of  the  North  half  of 
the  same  section,  how  would  the  part  sold  be  designated  ? 
What  would  be  the  designation  of  the  part  retained? 

CIRCUMFERENCE  AND  AREA  OF  A  CIRCLE. 

429.  Carefully  cut  out  of  cardboard  a  circle  having  a 
diameter  of  If  inches.  Make  a  dot  A  on  the  circumference, 
and  placing  the  dot  at  the  point  X  on  a  sheet  of  paper,  re- 
volve the  circle  until  the  dot  again  touches  the  paper  at  Y, 
which  is  marked.  Measure  the  distance  XY,  which  should 
be  5^  inches. 


430.    The  ratio  between  the   diameter   and   the   circum- 
ference is  the  ratio  between  1}  inches  and   5^   inches,  or 


21 8  Arithmetic 

7   to   22.     The  circumference   is,  therefore,  3f   times   the 
diameter.  * 

431.  To  determine  the  length  of  the  circumference  of  a 
circle  multiply  the  length  of  the  diameter  by  3^. 

Note.  Circumference  =  diameter  x  3.1416  is  more  accurate  when 
the  circle  is  very  large.     Unless  otherwise  specified,  use  3}. 

432.  Sight  Exercises. 

Find  the  circumferences  of  circles  having  diameters  as 
follows : 

1.  14  yd.  4.   35  rd.  7.   63  in. 

2.  28  ft.  5.    49  yd.  8.    42  mi. 

3.  21  in.  6.    70  rd.  9.    56  ft. 

Find  the  diameters  of  circles  having  circumferences  as 
follows : 

10.  220  in.  13.    66  yd.                  16.    110  mi. 

11.  44  rd.  14.    198  ft.                 17.    88  ft. 

12.  132  rd.  15.    176  rd.                 18.    154  in. 

Find  the  radii  of  circles  having  circumferences  as  follows  : 

19.  132  rd.  22.    44  rd.  25.    440  in. 

20.  88  yd.  23.    110  ft.  26.   220  rd. 

21.  176  mi.  24.    264  in.  27.    308  ft. 

Find  the  circumferences  of  circles  having  radii  as  follows : 

31.  7  ft.  34.    28  mi. 

32.  24^  yd.  35.    lOi- rd. 

33.  21  rd.  36.    31^  yd. 

433.  Draw  on  stiff  paper  a  circle  having  a  radius  of  3|- 
inches.  Divide  it  into  16  equal  parts  by  cuts  passing 
through  the  center.  Place  these  in  two  rows,  as  shown  in 
the  accompanying  diagram. 


28. 

31  in. 

29. 

35  ft. 

30. 

14  rd. 

Mensuration 


434.  By  increasing  the  number  of  pieces,  AD  gradually 
approaches  a  straight  line  11  inches  long,  one  half  the 
length  of  the  circumference ;  and  AC  gradually  approaches 
a  perpendicular  measuring  3i  inches,  one  half  the  length  of 
the  diameter. 

435.  When  the  number  of  pieces  becomes  indefinitely 
large,  the  figure  becomes  a  rectangle  11  inches  by  3|-  inches. 

436.  Substituting  tt  for  31,  the  ratio  between  the  circum- 
ference of  a  circle  and  its  diameter,  and  R  for  the  radius  of 
the  circle,  the  line  AD,  which  equals  one  half  of  the  circum- 
ference, will  be  ttB,  and  AB  will  be  R. 

The  area  of  a  circle,  therefore,  is  ttR-  ;  that  is,  the  square 
of  the  radius  multiplied  by  3i. 

Note.  The  Greek  letter  tt  (pronounced  pi)  is  used  in  formulas  to 
Indicate  the  ratio  of  the  length  of  the  diameter  to  that  of  the  circum- 
ference.    It  stands  for  3j,  or,  more  accurately,  3.1416. 

437.  TJie  numher  of  square  units  m  the  area  of  a  circle  is 
equal  to  the  square  of  the  nuviher  of  units  in  its  radius  multi- 
plied  by  3^. 

438.  Sight  Exercises. 

Find  the  areas  of  circles  with  dimensions  as  given  on  the 
following  iDage : 

Area  =  ^  circumference  x  |  diameter.     Area  =  S}B^. 
Use  either  formula. 


220 


Arithmetic 


1.  Eadius  3i  ft. 

2.  Diameter  7  yd. 

3.  Circumference  22  in. 

4.  Radius  1  rd. 

5.  Diameter  2  mi. 

439.   Written  Problems. 


6.  Circumference  64  ft. 

7.  Radius  2  yd. 

8.  Diameter  6  rd. 
9. 

10.  Radius  7  in. 


Circumference  18|^  rd. 


1.  If  the  diameter  of  a  wheel  of  a  wagon  is  4  feet,  how  far 
does  the  wagon  go  during  one  revolution  of  the  wheel  ? 
How  many  times  will  the  wheel  revolve  in  going  5280 
feet? 

2.  If  a  wagon  wheel  4  feet  in  diameter  revolves  420 
times  in  an  hour,  how  long  will  it  take  to  travel  1  mile  ? 

3.  How  many  revolutions  per  minute  are  made  by  a 
wagon  wheel  4  feet  in  diameter  when  the  wagon  travels  at 
the  rate  of  3  miles  per  hour  ? 

4.  Find  the  number  of  revolutions  per  minute  made  by 
a  locomotive  wheel  4  feet  in  diameter  when  the  train  is 
traveling  at  the  rate  of  a  mile  in  2  minutes. 

5.  How  many  feet  are  there  in  an  arc  of  60°  of  a  circle 
whose  radius  measures  6  feet? 

6.  Find  the  radius  of  a  circle  whose  area  is  154  square 
inches.     Find  the  diameter.     Find  the  circumference. 

7.  What  is  the  length  of  the  circumference  of  a  circle 
containing  an  area  of  616  square  inches  ? 

8.    A  horse  is  tied  at  M,  the  corner  of   a 

fence,    by  a   rope  20  feet  long.     How  many 

square   feet  of  surface  can  he  graze  over  if 

he  can  reach  one  foot  beyond  the  end  of  the 

rope  ? 

MX  =  MY  =  21  feet. 


Mensuration 


221 


9.    An  elliptical  flower  bed  measures  14  feet  on  the  axis 
AB   by   lOi   feet   on   axis    CD.     How 
many  square  feet  does  it  contain? 


Area: 


2  2 


10.    The    long   axis   of    an   elliptical 
pond  measures  21  yards,  and  the  short 
axis  9  yards  1  foot.     How  many  square  yards  are  there  in 
its  surface? 


PRISMS. 

440.  A  solid  is  a  portion  of  space  bounded  by  surfaces. 
When  the  side  surfaces  are  parallelograms,  and  the  ends  or 
bases  are  equal  parallel  polygons,  the  solid  is  called  a  prism. 

A  prism  may  be  triangular.,  rectangular.,  pentagonal,  etc.,  accord- 
ing to  the  number  of  sides  in  its  bases. 

441.  A  right  prism  is  one  whose  sides  are  rectangles. 
Fig.  1  shows  a  right,  square  prism. 


1 
I 
I 
I 
I 


Fig.  t. 


442.    The  altitude  of  a  prism  is  the  perpendicu- 
lar distance  between  the  planes 
of  its  bases.     AX,  Fig.  2,  is  the 
altitude  of  the  oblique  triangular 
prism . 

Note.    Unless  otherwise  speci- 


fied,   the  teiTn  prism   is   understood   to   mean    a 
right  prism. 


Fig.  2. 


443.  The  convex  surface  of  a  prism  is 
the  surface  exclusive  of  its  bases ;  the  entire  surface  includes 
the  surface  of  both  bases. 

444.  In  Fig.  3  (page  222)  is  represented  a  hollow  cardboard  prism 
opened  out  to  show  its  surface.  This  is  called  the  development  of 
a  solid.  AD  shows  the  height  of  the  prism,  ah  one  side  of  the  base 
and  ad  the  other. 


222 


Arithmetic 


445.    Sight  Exercises- 
Assuming   the    prism,   whose  development  is    shown   in 

Fig.  3,  to  be  8  inches  high,  and  the  base  to  be  4  inches  by 

2  inches : 

1.  Find  the  area  of  Bdfc. 

2.  Find  the  area  of  adfe. 

3.  Find  the  area  of  ahcd. 

4.  Find  the  area  of  ABCD. 

5.  Find  the  area  of  rectangle  1. 

6.  Find  the  area  of  rectangle  2. 

7.  Find  the  convex  surface. 

8.  Find  the  entire  surface. 

9.  What   is   the  length  of   the   base   of   the   rectangle 
forming  the  convex  surface  ? 

10.    What  is  the  length  of  the  perpendicular  of  the  rec- 
tangle forming  the  convex  surface  ? 


h 

c 

A                 a 

M 

d  B 

1 

2 

3 

4 

Z> 

e 

K 

./■  c 

h 
Fig.  3. 


446.  The  number  of  square  units  in  the  convex  surface  of  a 
right  prism  is  equal  to  the  ijroduct  of  the  7iumber  of  units  in 
the  perimeter  of  the  base  by  the  number  of  units  in  the  height. 

The  entire  surface  =  convex  surface  +  area  of  bases. 

447.  "Written  Exercises. 

1.  How  many  square  yards  in  the  convex  surface  of  a 
hexagonal  prism,  each  side  of  the  base  measuring  3  feet, 
and  the  height  being  10  feet  ? 

2.  The  base  of  a  prism  is  a  triangle  whose  sides  measure 
3  inches,  4  inches,  and  5  inches,  respectively.  Its  altitude 
is  12  inches.     How  many  square  inches  in  the  entire  surface  ? 

3.  The  entire  surface  of  a  square  prism  contains  112 
square  feet.  Each  side  of  the  base  measures  4  feet.  What 
is  the  convex  surface  of  the  prism  ?     Find  its  altitude. 


Mensuration  223 

4.  What  is  the  edge  of  a  cube  whose  entire  surface  con- 
tains 486  square  inches  ? 

5.  A  prism  6  feet  high  has  a  rectangular  base  2  feet 
8  inches  by  1  foot  6  inches.  How  many  square  feet  in  the 
entire  surface  ?  Wouhl  the  convex  surface  of  the  prism  be 
greater  or  less  if  the  base  were  a  square  containing  the 
same  area  as  the  rectangle  ? 

CYLINDERS. 

448.   A  cylinder,  Fig.  4,  is  a  solid  having  the  same  diam- 
eter throughout  and  two  equal  and   parallel  circles 
for  bases. 

Cylinders,  like  prisms,  are  either  right  or  oblique. 

The  developed  surface  of  the  cylinder  is  shown  in  Fig.  5.   yig    4 
The   line   AB  is   equal  in  length  to  the 
X  length  of  the  circumference  of  the  circle  forming 

^^r  the  base. 


449.   Sight  Exercises. 

Assuming  the  cylinder,  whose  develop- 
-jj    ment  is  shown  in  Fig.  5,  to  be  10  inches  high, 
and  the  diameter  of  the  base  7  inches,  find  : 
,-,,^   .    •  1.    The  length  of  AB. 

2.    The  area  of  ABDC.     • 

3.  The  area  of  the  upper  base, 

4.  The  area  of  the  lower  base. 

5.  The  convex  surface. 

6.  The  entire  surface. 

450.    Written  Exercises. 

1.  Find  the  convex  surface  of  a  cylinder  14  feet  high,  the 
diameter  of  the  base  being  3  feet.     Find  the  entire  surface. 

2.  At  36  cents  a  square  yard,  find  the  cost  of  polishing  the 
convex  surface  of  a  cylindrical  marble  column  12  feet  high 


224  Arithmetic 

and  11  feet  in  circumference.  What  will  it  cost  to  smooth 
otf  the  ends  at  9  cents  a  square  yard  ? 

3.  A  roller  is  6  feet  long  and  31-  feet  high.  How  many 
square  feet  of  ground  does  it  roll  at  each  revolution  ?  How 
many  revolutions  does  it  make  in  going  the  length  of  a 
square  field  containing  10  acres  ?  How  many  times  must 
it  cross  the  field  to  roll  the  whole  field  ?  How  many  miles 
must  it  travel,  exclusive  of  turns  ? 

«  4.  The  convex  surface  of  a  cylinder  6  feet  high  is  132 
square  feet.  Find  the  circumference  of  the  base.  Find  the 
radius  of  the  base.  Find  the  area  of  each  base.  Find  the 
entire  surface  of  the  cylinder. 

5.  The  entire  surface  of  a  cylinder  is  113^  square  inches, 
the  base  being  4  inches  in  diameter.  What  is  the  convex 
surface  of  the  cylinder  ?     What  is  its  altitude  ? 

VOLUME  OF  PRISM  AND  CYLINDER. 

451.  The  numher  of  cubic  units  in  the  volume  of  a  prism  or 
of  a  cylinder  is  equal  to  the  product  of  the  numher  of  square 
units  in  the  base  by  the  number  of  liiiear  units  in  the  altitude. 

452.  Written  Exercises. 

1.  What  is  the  volume  of  a  triangular  prism  whose  alti- 
tude is  9  feet,  the  sides  of  the  base  measuring  3,  4,  and  5 
feet,  respectively  ? 

2.  A  cylinder  is  12  feet  high  and  11  feet  in  circumference. 
How  many  cubic  feet  does  it  contain  ? 

3.  Find  the  dimensions  and  the  cubical  contents  of  the 
largest  square  prism  that  can  be  made  from  a  cylindrical 
wooden  column  12  feet  high  and  11  feet  in  circumference. 
How  many  cubic  feet  of  wood  will  be  removed  in  making 
the  change  ? 

4.  A  grindstone  3|  feet  in  diameter  and  4  inches  thick  has 
an  opening  through  the  center  4  inches  square.     How  many 


Mensuration 


225 


pounds    does    it    weigh    at    135     pounds     to     the     cubic 
foot? 

5.  How  many  square  inches  are  there  in  the  entire  sur- 
face of  a  cylinder  7  inches  in  diameter  and  7  inches  high  ? 
How  many  cubic  inches  in  its  volume  ? 

PYRAMIDS   AND   CONES. 

453.    A  pyramid  is  a  solid  whose  base  is  a  polygon,  and 
whose  sides  are  triangles  terminating  in  a  com- 
mon vertex. 


A  cone  is  a  solid  sloping  regularly  to 
its  vertex  from  a  circular  base. 


Fig.  6. 


455.    In   a  right  cone  or  a  right  pyramid  the 
vertex   is  directly  over  the  center  of  the  base, 
J-  the  sides  of  a  right  pyramid  being  isosceles  tri- 
angles.     In   a  regular  right  pyramid  these  tri- 
angles are  equal. 

456.  In  Fig.  8  is  shown  the  development  of  a  square 
pyramid,  AX  denoting  the  altitude  of  each  isosceles  triangle. 
This  is  called  the  slant  height  of  the  pyramid. 

457.  Sight  Exercises. 

Assuming  that  the  slant  height  AX  of  the  pyramid  in 
Fig.  8  is  12  inches   and  that 
DG  is  10  inches  : 

1.  Find  the  area  of  DEFG. 

2.  Find  the  area  of  ADG. 

3.  Find  the  area  of  AGH. 

4.  Find  the  area  of  ADC. 

5.  Find  the  area  of  ABC. 

6.  What  is  the  convex  sur- 
face of  the  pyramid  ?  Fig.  8. 


226  Arithmetic 

7.  What  is  the  entire  surface  ? 

8.  The  convex  surface  of  the  pyramid  is  equal  to  the  slant 
height  multiplied  by  what  ? 

9.  What  is  the  perimeter  of  the  base  DEFG  ? 

10.  If  ^IX  is  12  inches  and  XD  is  5  inches,  what  is  the 
length  of  the  hypotenuse  AD  ? 

458.  TJie  number  of  square  inches  rn  the  convex  surface  of  a 
2)yramid  (or  of  a  cone)  is  equal  to  one  half  the  product  of  the 
number  of  inches  in  the  perimeter  (or  the  circumference)  by  the 
number  in  the  slant  height, 

459.  "Written  Exercises.  ♦ 

1.  Find  the  number  of  square  feet  in  the  convex  surface 
of  a  hexagonal  pyramid,  each  side  of  the  base  measuring  1 
foot  6  inches,  and  the  slant  height  being  4  feet  4  inches. 

2.  What  is  the  entire  surface  of  a  square  pyramid  whose 
slant  height  is  3  yards  1  foot,  each  side  of  the  base  being  27 
inches  ? 

3.  What  is  the  convex  surface  of  a  cone  whose  slant 
height  is  14  inches,  the  circumference  of  the  base  being  22 
inches  ? 

Note.  The  development  of  the  convex  surface  of  a  cone  is  a  sector, 
the  number  of  square  units  in  tlie  area  being  equal  to  the  product  of 
the  number  of  linear  units  in  the  arc  by  the  number  in  the  radius. 

4.  The  altitude  IIK  of  a  cone.  Fig.  7,  is  24  inches  and 
the  diameter  of  the  base  is  14  inches.  Find  the  slaijt 
height  HI.     Find  the  convex  surface  of  the  cone. 


HI=  VHK^  +  ^/^  =  \/242  +  7  ^ 

5.  The  steeple  of  a  church  is  in  the  form  of  an  octagonal 
pyramid,  each  side  of  the  base  measuring  6  feet.  How 
many  square  yards  are  there  in  its  convex  surface,  if  the 
slant  height  is  42  feet  ? 


Mensuration 


227 


VOLUME  OF  PYRAMID  AND  CONE. 

460.  From  a  semicircular  piece  of  stiff  paper,  Fig.  9, 
make  a  hollow  cone,  Fig.  10.  '  With  a  rectangular  piece  of 
stiff  paper,  Fig.  11,  make  a  hollow 
cylinder.  Fig.  12 ;  the  width  of  the 
paper,  MN,  being  exactly  the  altitude 
of  the  cone,  AB. 

Use  the  cone  to  fill  the  cylinder  with 
sand,  holding  the  bottom  of  the  cylinder 
against  a  plate  to  prevent  the  escape  of  the   sand 
many  cones  of  sand  will  the  cylinder  contain  ? 


Fig.  9. 


How 


Fig.  11. 


Fig.  12. 


461.  The  number  of  cubic  units  in  the  volume  of  a  pyramid 
or  of  a  cone  is  equal  to  one-third  of  the  jv'oduct  of  the  number 
of  square  units  in  the  base  by  the  number  of  linear  units  in 
the  altitude. 

462.  Written  Exercise's. 

1.  Find  the  volume  of  a  square  pyramid  12  feet  high, 
each  side  of  the  base  measuring  5^  feet. 

2.  A  cylindrical  can  is  7  inches  in  diameter  and  7  inches 
high.  A  cone  of  the  same  dimensions  is  placed  in  the 
can.  How  many  cubic  inches  of  water  will  be  required  to 
fill  the  remaining  space  ? 

3.  A  p3'ramid  12  inches  high  has  a  base  10  inches  square. 
If  a  pyramid  6  inches  high  is  cut  from  the  top  by  a  plane 
parallel  to  the  base,  what  will  be  the  dimensions  of  the  base 
of  the  small  pyramid.  How  does  the  volume  of  the  small 
pyramid  compare  with  the  volume  of  the  original  pyramid? 


228 


Arithmetic 


4.  At  1^  cubic  feet  to  the  bushel,  find  the  number  of 
bushels  that  will  be  contained  in  a  conical  pile  of  wheat  22 
feet  in  circumference  at  the  base  and  7  feet  high. 

5.  Find  the  number  of  bushels  in  a  pile  of  grain  in  the 
corner  of  a  granary,  if  the  pile  is  7  feet  high,  each  point  of 
the  base  being  3^  feet  from  the  corner. 

What  part  of  a  cone  is  formed  by  the  pile  ? 


THE   SPHERE. 

463.  A  sphere  is  a  solid,  all  points  on  the  surface  of  which 
are  equally  distant  from  its  center. 

464.  The  sphere,  the  right  cylinder,  and  the  right  cone 
are  called  solids  of  revolution. 

465.  The  cone  is  formed  by  the  revolution  of  the  right 
triangle  ABC,  Fig.  13,  on  the  perpendicular 
AB  as  its  axis,  the  point  C  of  the  base  BC 
of  the  triangle  describing  the  circumference 
DNCM,  which  bounds  the  base  of  the  cone. 

466.    In  the  same  way,  the  cylinder,  Fig.  14, 
is   formed   by    the   revolution    of     ^^^^-?— -^ 
the  rectangle  STQR,  on  the  axis  SR,  the  side  '^^^^ZlU-^ 
EQ  forming  the  base,  and  TQ  the  curved  sur- 
face. 

467.  The  sphere  is  ^' 
formed  by  the  revolution 
of  a  semicircle,  AXB,  Fig.  15,  on  the 
diameter  AB  as  the  axis.  The  radius 
-^  CX,  perpendicular  to  the  axis,  de- 
scribes a  great  circle  of  the  sphere. 
The  half  chords  Ma  and  Nh,  perpen- 
dicular to  the  axis,  describe  small 
circles. 


Mensuration  229 

468.  A  cutting  plane  passed  through,  a  sphere  in  any 
direction  will  leave  the  cut  surface  a  circle.  If  the  plane 
passes  through  the  center  of  the  sphere,  the  cut  surface 
will  be  a  great  circle. 

469.  The  radius  of  a  sphere  is  a  line  from  the  center  of 
a  sphere  to  its  surface.  CA,  Ca,  CX,  and  CB,  Fig.  15,  are 
radii. 

470.  Drive  a  tack  into  the  center  of  the  curved  surface 
of  the  half  of  a  croquet  ball  or  other  wooden  hemisphere. 
Starting  at  the  tack,  wind  a  stout  cord  about  the  curved 
surface  in  the  way  a  top  is  wound.  When  this  surface  is 
entirely  covered,  cut  the  cord.  Drive  another  tack  into  the 
center  of  the  plane  surface,  and  carefully  wind  the  same 
cord  on  the  plane  surface.  When  this  task  is  finished,  it 
will  be  found  that  just  one  half  of  the  piece  of  cord  is  re- 
quired. The  area  of  the  curved  surface  is,  therefore,  twice 
the  area  of  the  plane  surface. 

471.  The  area  of  the  plane  surface  of  the  hemisphere  is 
the  area  of  a  great  circle  of  the  sphere.  The  area  of  the 
convex  surface  of  the  hemisphere  is,  therefore,  equal  to  the 
area  of  two  great  circles ;  and  the  area  of  the  entire  surface 
of  the  sphere  is  equal  to  the  area  of  four  great  circles. 

472.  This  may  be  expressed  as  follows : 

Surface  of  sphere  =  4  ttR-  ; 
and,  since  the  square  of  the  diameter  is  4  times  the  square 
of  the  radius,  the  formula  may  be  expressed  thus : 

Surface  of  sphere  =  ttD'-. 

473.  The  number  of  square  units  in  the  surface  of  a  sphere 
is  equal  to  the  square  of  the  number  of  units  in  the  diameter 
multiplied  by  34. 


230  Arithmetic 

VOLUME  OF  A  SPHERE. 

474.  Mold  carefully  a  lump  of  clay  into  a  good-sized 
ball.  Make  a  stout  paper  cylinder  having  its  altitude  and 
the  diameter  of  its  base  equal  to  the  diameter  of  the  ball. 
Make  a  paper  cone  of  the  same  base  and  altitude  as  the 
cylinder.  Place  the  ball  in  the  cylinder  and  pour  into  the 
cylinder  the  cone  full  of  water,  which  exactly  fills  it. 

475.  The  volume  of  the  sphere  is  equal  to  the  volume  of 
the  cylinder  less  the  volume  of  the  cone. 

Volume  of  cylinder  =  circle  X  altitude. 
Volume  of  cone        =  circle  x  ^  altitude. 
Volume  of  sx3here     =  circle  x  |  altitude. 

The  area  of  the  circle  being  ttR-,  the  volume  of  the  sphere 
=  TT^^X  I  i)  or  ttR'  X  ^  B,  which  equals  |  ttB^.  Substitut- 
ing I  D^  for  B\  we  have,  area  =  |-  tt  x  ^  D^,  or  i  ttD". 

476.  The  number  of  cubic  units  in  the  volume  of  a  sphere  is 
equal  to  one  sixth  of  the  product  of  the  number  of  units  in 
the  cube  of  the  diameter  by  3\. 

477.  Note.  If  a  sphere,  a  cylinder,  and  a  cone  of  the  same  material 
are  available,  the  weight  of  the  cylinder  should  equal  the  weight  of  the 
other  two,  provided  the  altitude  of  the  cylinder  and  of  the  cone,  and 
the  diameter  of  the  base  of  each,  are  equal  to  the  diameter  of  the 
sphere.  If  the  water  that  fills  a  rubber  ball  is  squeezed  into  a  cone 
having  the  same  diameter  as  the  inner  diameter  of  the  ball  and  having 
its  altitude  equal  to  the  diameter,  it  should  fill  the  cone  twice. 

478.  Written  Exercises. 

1.  Eind  the  surface  of  a  sphere  whose  diameter  is 
7  inches. 

2.  What  is  the  entire  surface  of  a  cylinder  7  inches 
high  and  7  inches  in  diameter  ? 

3.  Find  the  volume  of  a  sphere  whose  diameter  is 
7  inches. 


Miscellaneous  Problems  231 

4.  Find  the  volume  of  a  cone  7  inches  high  and  7  inches 
in  diameter  at  the  base.  Find  the  volume  of  a  cylinder 
7  inches  high  and  7  inches  in  diameter.  Find  the  difference 
between  the  two  volumes. 

5.  If  a  sphere  7  inches  in  diameter  is  placed  in  a  hollow 
7-inch  cube,  how  many  cubic  inches  of  water  will  the  cube 
then  hold  ? 

6.  A  hemispherical  bowl  of  iron  is  7  inches  in  diameter. 
How  many  cubic  inches  of  iron  does  it  contain  if  the  iron  is 
1  inch  thick  ? 

Find  the  difference  between  the  volume  of  a  7-inch  hemisphere 
and  that  of  a  6-inch  hemisphere. 

7.  Find  the  cost  of  gilding  a  ball  14  inches  in  diameter 
at  36  cents  per  square  foot. 

8.  If  cast  iron  weighs  7  times  as  much  as  water,  find 
the  weight  of  a  solid  iron  ball  6  inches  in  diameter,  the 
weight  of  a  cubic  foot  of  water  being  1000  ounces. 

9.  What  is  the  ratio  between  the  volume  of  a  3-inch 
sphere  and  that  of  a  6-inch  sphere  ? 

10.  What  is  the  largest  sphere  that  can  be  cut  from  a 
cubical  block  of  granite  12  inches  on  each  edge  ?  What 
decimal  of  a  cubic  foot  of  material  will  be  cut  away  ? 

MISCELLANEOUS  PROBLEMS. 
479.    Miscellaneous  Oral  Problems. 

1.  How  many  rods  of  fence  will  be  required  to  inclose 
a  square  10-acre  field  ? 

2.  A  contractor  has  30  days  in  which  to  finish  a  piece 
of  work.  His  force  of  60  men  would  require  50  days  to 
complete  it.     How  many  additional  men  must  he  employ  ? 

3.  What  are  the  cubical  contents  of  a  cellar  20  feet  long, 
15  feet  wide,  and  10  feet  deep  ? 


232 


Arithmetic 


4.  If  5  sheep  are  worth  $  20,  how  many  calves  at  $  3 
each  must  be  given  in  exchange  for  12  sheep? 

5.  Three  fifths  of  a  yard  of  ribbon  costs  7^  cents.  How 
many  yards  can  be  bought  for  $  5  ? 

6.  In  dividing  300  firecrackers  among  some  boys  there 
are  6  left  after  each  boy  receives  14.  How  many  boys  are 
there? 

7.  Find  the  cost  of  16  bushels  of  wheat  at  99^-  cents  per 
bushel. 

8.  Eight  men  cut  20  cords  of  wood  per  day.  How  long 
will  it  take  6  men  to  cut  80  cords  ? 

9.  The  contents  of  a  rectangular  box  are  136  cubic  feet. 
Its  length  is  6  feet  and  its  depth  is  4  feet.  Find  its  width 
in  feet  and  inches. 

10.  William  and  Thomas  can  do  in  8  days  a  piece  of 
work  which  the  latter  can  do  alone  in  12  days.  How  long 
would  it  take  William  to  do  it  alone  ? 

11.  How  many  rectangular  blocks  each  2  inches  by  3 
inches  by  4  inches  will  be  required  to  fill  a  box  measuring 
2  feet  by  3  feet  by  4  feet  ? 

12.  When  eggs  are  sold  at  15  cents  per  dozen,  a  profit  of 
^  the  cost  is  made.     What  did  they  cost  ? 

13.  Find  the  solid  contents  of  a  cube,  the  area  of  each 
face  of  which  is  400  square  inches. 

14.  What  will  be  the  value  of  a  pile  of  wood  4  feet  wide, 
12  feet  long,  and  4  feet  high,  at  $4.50  per  cord? 

15.  Find  the  cost  of  49  pounds  of  tea  at  41  cents  per 
pound. 

16.  Each  face  of  a  cube  contains  81  square  inches.  Find 
its  volume. 

17.  What  is  the  hypotenuse  of  a  right  triangle  if  the 
other  sides  measure  6  and  8  feet,  respectively  ? 


Miscellaneous  Problems  2^2 

18.  At  18  cents  per  square  yard,  what  will  be  the  cost  of 
a  blackboard  10  feet  long,  4  feet  wide  ? 

19.  How  many  acres  in  a  field  40  rods  long,  40  rods  wide  ? 

20.  What  is  the  side  of  a  square  field  containing  64  times 
as  many  acres  as  a  square  field  40  rods  on  a  side  ? 

21.  How  many  3-inch  cubes  can  be  placed  in  a  box  12 
inches  long,  9  inches  wide,  6  inches  high  ?  How  many  1-inch 
cubes  can  be  placed  in  the  same  box  ? 

22.  If  J  bushel  occupies  a  cubic  foot,  how  many  bushels 
will  there  be  in  a  wagon  body  that  holds  just  a  cubic  yard? 

23.  How  many  square  rods  in  a  field  65  rods  long,  65  rods 
wide? 

24.  Eind  the  number  of  board  feet  in  a  stick  of  timber  12 
feet  long,  8  inches  wide,  3  inches  thick. 

25.  If  there  are  27  bricks  in  a  cubic  foot,  and  221  bricks 
with  the  mortar  make  a  cubic  foot  of  wall,  what  part  of  the 
wall  is  mortar  ? 

26.  Making  no  allowance  for  openings,  how  many  rolls 
of  paper  will  be  needed  for  the  walls  of  a  room  20  feet  long, 
15  feet  wide,  and  10  feet  high,  3  rolls  being  required  for  100 
square  feet  ? 

27.  At  1^  cubic  feet  to  the  bushel,  how  many  bushels  can 
be  placed  in  a  bin  10  feet  long,  8  feet  wide,  and  4  feet  deep  ? 

28.  How  many  pounds  of  flour  are  there  in  25  barrels 
each  containing  196  pounds  ? 

29.  How  many  bricks  are  there  in  a  cubic  foot,  if  a  brick 
measures  f  foot  by  i  foot  by  i  foot? 

30.  Assuming  that  wrought  iron  is  8  times  as  heavy  as 
water,  what  is  the  weight  of  a  cubic  foot  of  iron,  a  cubic 
foot  of  water  weighing  1000  ounces  ? 

31.  A  room  is  20  feet  long,  15  feet  wide,  9  feet  high. 
How  many  square  yards  in  the  walls  ? 


234  Arithmetic 

32.  Two  partners  contribute  $  400  and  $  500,  respectively. 
How  will  they  divide  profits  of  $  180  ? 

33.  How  many  inch  cubes  can  be  placed  in  a  box  8  inches 
long,  5  inches  wide,  and  3  inches  deep  ? 

34.  Find  the  cost  of  60  yards  of  muslin  at  ll^-  cents  per 
yard. 

35.  If  each  shingle  is  laid  so  that  ^  foot  by  ^  foot  is  ex- 
posed to  the  weather,  how  many  shingles  will  be  needed 
for  100  square  feet,  not  counting  waste  ? 

36.  How  many  pounds  equal  1000  ounces  ? 

37.  At  27  cents  per  square  yard,  what  will  it  cost  to 
plaster  the  ceiling  of  a  room  20  feet  long,  15  feet  wide  ? 

38.  A  bushel  of  corn  in  the  ear  measures  21  cubic  feet. 
How  many  bushels  will  occupy  3300  cubic  feet  ? 

39.  How  many  yards  of  carpet  f  yard  wide  will  be  needed 
to  cover  a  floor  15  feet  by  9  feet  ? 

40.  When  a  man  6  feet  tall  casts  a  shadow  of  4  feet,  the 
shadow  of  a  steeple  is  48  feet.     How  high  is  the  steeple  ? 

41.  How  many  board  feet  are  there  in  10  boards  each  12 
feet  long,  8  inches  wide,  2  inches  thick  ? 

42.  If  a  1-foot  iron  cube  weighs  500  pounds,  what  will  be 
the  weight  of  a  2- foot  cube  ? 

43.  What  is  the  smallest  whole  number  by  which  24  must 
be  multiplied  to  produce  a  perfect  square  ? 

44.  A  man  of  48  is  3  times  as  old  as  his  son.  What  will 
be  the  ratio  of  their  respective  ages  in  16  years  ? 

45.  What  is  the  cost  of  an  acre  of  land  at  the  rate  of 
$21.75  for  f  acre? 

46.  If  taxes  are  $  1.50  per  $  1000,  what  must  be  paid  on 
property  worth  $  4800  ? 

47.  What  is  the  smallest  number  that  will  have  a  re- 
mainder of  1  when  divided  by  3,  by  4,  or  by  5  ? 


Miscellaneous  Problems  2^^ 

48.  A  can  do  a  piece  of  work  in  4  days,  B  can  do  it  in 
6  days.  How  many  days  will  it  require  for  both  to  do  the 
work  ? 

49.  A  man  owing  $  1500  has  only  f  1000.  How  much 
should  M  receive  to  whom  $  240  is  due  ? 

50.  Divide  60  into  3  numbers  proportional  to  3,  4,  and  5. 

51.  Philadelphia  has  the  time  of  75°  west  longitude,  and 
Berlin  that  of  15°  east  longitude.  AVhat  is  the  time  at 
Philadel]3hia  when  it  is  3  p.m.  at  Berlin  ? 

52.  How  many  acres  are  there  in  a  field  120  rods  long, 
80  rods  wide  ? 

53.  $  6.24 -- 1.04  =  ? 

54.  Three  25ths  is  what  part  of  three  5ths  ? 

55.  How  many  blocks  1  inch  x  1  inch  x  1  inch  will  be 
required  to  make  a  pile  1  foot  x  1  foot  x  1  foot  ? 

56.  Find  the  solid  contents  of  a  cube,  the  area  of  one 
face  being  100  square  inches. 

57.  A  bale  of  hay  will  last  a  horse  3  weeks  or  a  calf  6 
weeks.     How  long  will  it  last  both  ? 

58.  What  is  the  value  of  a  pile  of  4-foot  wood  16  feet 
long,  4  feet  high,  at  $  4.50  per  cord  ? 

59.  How  many  board  feet  in  a  piece  of  timber  18  feet 
long  and  10  inches  square  ? 

60.  Find  the  convex  surface  of  a  pyramid  whose  base  is 
5  feet  square  and  whose  slant  height  is  6  feet. 

61.  What  will  be  the  cost  at  $  1  per  1000  cubic  feet  for 
the  gas  burned  during  November,  if  5  lights,  each  con- 
suming 12  cubic  feet  per  hour,  are  used  per  night  for  an 
average  of  4  hours  each  ? 

62.  What  does  a  dealer  receive  for  a  2240-pound  ton  of 
coal  sold  for  $  5  per  ton  of  2000  pounds  ? 


236  Arithmetic 

63.  How  many  feet  of  board  will  be  required  for  a  tight 
fence  6  feet  high  inclosing  a  lot  25  feet  x  100  feet  ? 

64.  How  long  will  it  take  a  train  to  go  30  miles  at  the 
rate  of  50  miles  an  hour  ? 

65.  By  selling  a  house  for  $  3200,  the  owner  lost  \  of  its 
cost.     What  did  he  pay  for  it  ? 

66.  What  is  the  ratio  of  the  area  of  a  20-foot  square  to 
that  of  a  30-foot  square  ? 

67.  Four  fifths  of  what  number  equals  96  ? 

480.    Miscellaneous  Written  Problems. 

1.  A,  lot  44  feet  by  110  feet  sells  for  $850.  What 
would  an  acre  cost  at  the  same  rate? 

2.  Find  the  cost  of  7  one-inch  boards  16  feet  long,  8 
inches  wide,  at  $  23  per  M. 

3.  If  a  plow  turns  a  furrow  10  inches  wide,  what  is  the 
total  length  of  all  the  furrows  turned  in  plowing  a  10-acre 
field? 

4.  How  many  steps  of  2  feet  4  inches  each  will  it 
take  to  measure  a  mile  ? ' 

5.  A  floor  20  feet  4  inches  by  16  feet  8  inclies  is  laid 
with  tiles  4  inches  square.     How  many  tiles  are  used  ? 

6.  A  man  left  $3600  by  will  to  be  distributed  among 
three  servants  in  proportion  to  their  times  of  service,  which 
were  7^,  lOi,  and  15|  years,  respectively.  What  was  the 
share  of  each  ? 

7.  Show  how  a  concrete  quotient  must  be  obtained.  If 
the  multiplicand  be  concrete,  what  must  be  the  unit  of  the 
product  ?  Explain  why  a  concrete  number  cannot  be  sepa- 
rated into  two  like  factors. 

8.  A  ladder  50  feet  long  is  so  placed  in  the  street  that 
without  being  moved  at  the  foot,  it  will  reach  a  window  on 


Miscellaneous  Problems  237 

one  side  40  feet  and  on  the  other  side  48   feet  from  the 
ground.     How  wide  is  the  street  ? 

9.  In  what  time  will  8  masons  build  a  wall  84  feet  long, 
working  10  hours  a  day,  if  12  masons  build  a  wall  96  feet 
long  in  8  days,  working  8  hours  a  day  ? 

10.  Find  the  cost  of  the  lumber  required  for  a  floor  20 
feet  long,  18  feet  wide,  and  2  inches  thick,  at  $  15  per  M. 

11.  Three  pipes  will  fill  a  cistern  in  4  hours.  The 
first  would  require  12  hours  to  fill  it  and  the  second  8 
hours.     How  long  would  it  take  the  third  to  fill  it  ? 

12.  A  pile  of  wood  12  feet  long,  6  feet  high,  and  4  feet 
wide  is  sold  for  S  11.25.  At  what  price  per  cord  was  it 
sold? 

13.  At  the  time  a  staff  6  feet  above  the  ground  casts  a 
shadow  measuring  8  feet,  the  shadow  of  a  steeple  measures 
144  feet.     How  high  is  the  steeple  ? 

14.  Divide  $2380  between  A  and  B  so  that  |  of  A's 
share  will  equal  B's. 

15.  A  merchant  sold  |  of  a  quantity  of  cloth  at  4  of  its 
cost,  and  the  remainder  at  cost,  thereby  gaining  $  7.20. 
What  did  the  cloth  cost  ? 

16.  If  24  men  in  5  days  can  build  a  wall  72  rods  long, 
how  many  rods  of  wall  can  15  men.  build  in  6  days  ? 

17.  How  many  cubic  feet  of  stone  will  be  required  to 
build  a  wall  2  feet  thick  and  4  feet  high  on  the  outside  of  a 
plot  20  feet  square,  making  no  allowance  for  mortar? 

18.  A  creamery  buys  1800  quarts  milk  per  day  at  a  cost 
of  2)^  per  quart.  From  100  quarts  of  milk  are  obtained  7 
pounds  butter,  which  sells  at  25^  per  pound,  12  quarts  butter- 
milk sold  at  3^  per  gallon,  and  84  quarts  skimmed  milk  sold 
at  5^  per  gallon.  The  daih^  expenses  for  labor,  etc.,  are 
$7.50.     What  is  the  daily  profit? 


238  Arithmetic 

19.  Divide  $10,450  into  two  parts  having  a  ratio  of  2 
to  3. 

20.  How  many  rods  of  fence  will  be  required  to  inclose  a 
field  in  the  form  of  a  right-angled  triangle  containing  40 
acres,  the  base  measuring  128  rods  ? 

21.  A  dairyman  has  10  cows  averaging  840  pounds  each. 
How  many  tons  of  hay  will  they  consume  in  a  year,  if  each 
receives  daily  -^^  of  its  weight  of  hay  ? 

22.  A  tank  can  be  filled  by  pipe  A  in  12  hours  and  pipe 
^  in  9  hours.  Pipe  C  will  empty  it  in  8  hours.  How  long 
will  it  require  for  A  and  B  to  fill  it,  if  C  is  closed  ?  If  all 
three  are  open  ?  How  long  will  it  take  to  empty  it  when 
full,  if  A  and  C  are  open  ?     If  B  and  C  are  open  ? 

23.  What  is  the  value  of  a  pile  of  wood  12  feet  long,  4 
feet  wide,  and  8^  feet  high,  at  $  4.50  per  cord  ? 

24.  A  cow  gives  milk  for  300  days  per  year,  yielding  16 
quarts  daily  for  35  days,  12  quarts  daily  for  65  days,  and 
5  quarts  daily  for  the  remainder  of  the  time.  If  one  half 
the  milk  is  sold  at  12^  per  gallon  and  the  other  half  at  16;^ 
per  gallon,  how  much  is  received  for  the  milk  during  the 
year  ? 

25.  How  many  miles,  exclusive  of  turns,  would  a  team 
be  required  to  travel  to  cut  10  acres  of  grain  with  a .  reaping 
machine,  if  the  machine  cuts  a  strip  66  inches  wide  ? 

Note.     An  acre  is  66  inches  wide  by  how  many  miles  long  ? 

26.  A  farmer  had  320  bushels  of  potatoes  to  sell.  He 
sold  one  half  of  them  in  "the 'fall  at  60^  per  bushel.  The 
other  half  he  sold  in  the  spring  by  weight,  after  they  had 
lost  "2^  of  the  original  weight,  at  75^  per  bushel.  How 
much  more  did  he  receive  for  those  sold  in  the  spring  ? 

27.  An  exporter  bought  through  an  agent  1500  tierces  of 
lard  at  f  7.111  per  tierce.  The  agent  charged  him  2^^  per 
tierce  for  buying.  What  was  the  total  cost  of  the  lard,  and 
the  price  per  pound,  a  tierce  containing  340  pounds  ? 


Miscellaneous  Problems  239 

28.  At  $20  per  M,  what  will  be  the  cost  of  a  piece  of 
timber  30  feet  long,  14  inches  square,  and  a  plank  20  feet 
long,  8  inches  wide,  2^  inches  thick  ? 

29.  A  cistern  which  holds  100  gallons  can  be  filled  from 
a  pipe  in  25  minutes  and  emptied  by  a  waste  pipe  in  45 
minutes.  If  both  are  opened  together,  how  long  will  it  take 
to  fill  the  cistern,  and  how  much  water  will  have  been  wasted 
by  that  time  ? 

30.  An  agent  sold  for  a  dealer  600  bales  of  cotton, 
averaging  500  pounds  per  bale,  at  11.04;^  per  pound.  How 
much  did  the  dealer  receive  for  the  cotton,  if  the  agent 
deducted  $  5  per  100  bales  for  selling  it  ? 

31.  A  fast  train  starting  at  8 :  30  a.m.  reaches  its  desti- 
nation at  4:45  P.M.,  making  3  stops  of  3  minutes  each. 
Find  the  average  speed  per  hour  of  running  time,  the  dis- 
tance being  440  miles. 

32.  If  a  team  makes  a  furrow  10  inches  wide  and  9.9 
miles  long  in  a  day,  what  part  of  an  acre  is  contained  in  the 
furrow  ? 

33.  If  71  gallons  of  w^ater  occupy  one  cubic  foot,  how 
many  gallons  of  water  would  a  cylindrical  boiler  contain 
whose  diameter  is  24  inches  and  whose  height  is  7  feet  ? 

34.  A  freight  car  30  feet  long,  9  feet  wide,  is  filled  with 
wheat  to  a  depth  of  8  feet.  How  many  bushels  of  wheat 
does  the  car  contain,  if  one  bushel  of  grain  occupies  IJ  cubic 
feet?  Find  the  weight  of  the  wheat  at  60  pounds  to  the 
bushel. 

35.  If  24  men  each  working  10  hours  a  day  do  two-thirds 
of  a  piece  of  work  in  28  days,  how  long  should  it  take  20 
men  working  8  hours  a  day  to  do  the  whole  work  ? 

36.  How  many  acres  are  there  in  a  rectangular  field  402 
rods  2  feet  long  and  120  yards  wide? 

37.  A  pile  of  wheat  5  feet  high  lies  in  the  corner  of  a 
barn,  each  point  of  the  base .  being  7  feet  from  the  corner. 


240  Arithmetic 

How  many  bushels  are  there  in  the  pile,  if  a  bushel  equals 
1 J  cubic  feet  ? 

One  fourth  of  a  cone  5  feet  high,  radius  7  feet. 

38.  A  and  B  were  doing  a  piece  of  work  which  both 
together  could  finish  in  15  days.  After  working  6  days,  A 
leaves,  and  B  completes  it  in  36  days.  How  many  days 
would  B  require  to  do  the  whole  work  ? 

39.  The  distance  between  two  towns  is  1584  kilometers. 
What  is  the  distance  in  miles  ? 

1  kilometer  =  1000  meters.     1  meter  =  39.37  inches. 

40.  A  map  representing  an  area  of  72,000  square  miles 
measures  12  inches  by  15  inches.  On  what  scale  is  it 
drawn  ? 

41.  A  farmer  hired  two  men  and  two  boys  to  harvest 
sugar  beets,  paying  them  $91.50  for  their  work.  If  they 
worked  221  days,  how  much  did  each  receive  per  day,  the 
boys  being  paid  one  half  as  much  as  the  men  ? 

42.  Write  one  thousand  forty-five  and  nine  millionths. 
Write  seven  and  nine  ten-thousandths. 

43.  How  much  wheat  will  it  take  to  seed  a  field  76  rods 
wide  by  105  rods  long,  at  the  rate  of  1^  bushels  to  the  acre  ? 

44.  Mr.  Lally  buys  3  horses  from  Mr.  Tully  at  $125.  In 
payment  he  delivers  to  the  latter  10  loads  of  hay  averaging 
1875  pounds  each,  for  which  he  receives  credit  at  the  rate  of 
$12  per  ton,  and  5  jars  of  butter  averaging  9  pounds  each, 
for  which  he  is  credited  at  the  rate  of  21^  per  pound.  Make 
out  Mr.  Tully's  statement  of  the  account. 

45.  A  cistern  4  meters  by  25  decimeters  by  2  meters  is 
supplied  by  a  pipe  which  discharges  50  liters  per  minute. 
How  long  will  it  take  to  fill  the  cistern  ? 

46.  The  foot  of  a  ladder  is  32  feet  from  the  base  of  a 
building  and  the  top  of  the  ladder  just  reaches  a  window 
60  feet  from  the  ground.     How  long  is  the  ladder  ? 


Miscellaneous  Problems  241 

47.  If  a  company  of  200  men  consumes  35  barrels  of 
flour  in  7  weeks,  how  long  will  it  take  400  men  to  consume 
40  barrels  ? 

48.  How  many  board  feet  in  6  pieces  of  timber,  each  20 
feet  long  and  10  inches  square  ? 

49.  The  sum  of  three  numbers  is  940.     The  first  equals 

1  of  the  second  and  the  second  equals  y^^  of  the  third.     Find 
the  numbers. 

50.  In  a  tank  11  feet  by  7  feet  the  oil  was  4  feet  deep. 
How  many  gallons  were  drawn  when  the  oil  measured  2 
feet  9  inches  in  depth  ? 

51.  A  country  is  480  miles  long  and  360  miles  wide. 
Give  the  dimensions  of  the  paper  on  which  is  drawn  the 
map  on  the  scale  of  ^  inch  to  the  mile,  with  a  border  of 

2  inches. 

52.  The  population  of  a  certain  city  is  i  greater  each 
year  since  1902  than  it  was  the  previous  year.  In  1905  the 
population  was  108,000.  What  was  the  population  in  1902  ? 
In  1907  ? 

53.  What  are  the  dimensions  of  a  rectangle  whose  area  is 
2187  square  feet  and  whose  length  is  three  times  its  breadth? 

Note.     Make  a  diagram.     Divide  the  rectangle  into  equal  squares. 

54.  Divide  $269.50  among  four  persons  in  the  ratio  of 
30,  20,  15,  and  12.     In  the  ratio  of  ^,  i  \,  f 

55.  Simplifyd  xtVx4f)-(li  +  4f). 

56.  Find  the  value  of  the  following: 

(0.139  X  28)  +  (42  X  0.002)  +  (6  x  0.004)  -  (0.05  x  20). 

57.  A  pile  of  wood  containing  67^  cords  is  270  feet  long 
and  4  feet  wide.     How  high  is  it  ? 

58.  If  from  18J  yards  of  cloth  4|i  yards  are  cut,  what 
fraction  of  the  piece  is  left  ? 


242  Arithmetic 

59.  A  square  court  is  paved  with  48,841  stones,  each  one 
foot  square.     Find  the  dimensions  of  the  court. 

60.  A  can  do  i  of  a  piece  of  work  in  a  day  and  B  can  do 
1  of  it  in  a  day.  They  work  together  and  receive  $  6  for 
the  job.  How  long  does  it  take  them  to  do  the  work,  and 
how  much  should  each  receive  ? 

61.  Find  the  cost  of  12  planks,  each  10  feet  long,  12 
inches   wide,  and  3  inches  thick,  at  $  75  per  M-. 

62.  Find  the  result  of  the  following : 

[(72.2-10)  -  2]  -  (0.5  -  1.6)  +  [2.125-  (1.75  -  .05)]. 

63.  What  is  the  cost  of  a  board  24  feet  long,  23  inches 
wide  at  one  end  and  17  inches  wide  at  the  other,  and  1^ 
inches  thick,  at  $  30  per  M  ? 

64.  Extract  the  cube  root  of  15.625. 

65.  Reduce  ^x  mile  to  rods,  yards,  and  feet. 

66.  A  farmer  kept  450  bushels  of  potatoes  during  tjie 
winter,  losing  .1  by  decay.  He  sold  the  remainder  in  the 
spring  for  80  cents  a  bushel,  receiving  $54  more  than  he 
could  have  obtained  in  the  fall.  What  was  the  price  in  the 
fall  ? 

67.  After  spending  ^  of  his  money,  Jq-  of  the  remainder, 
and  y^Q  of  what  he  then  had,  a  man  has  $  142.56.  What  had 
he  at  first  ? 

68.  How  many  barrels  of  water,  of  31^  gallons  each,  will 
fall  on  a  garden  5  rods  by  6  rods,  during  a  shower  in  which 
the  fall  of  rain  is  three  fourths  of  an  inch  ? 

69.  A  circular  park  has  a  road  around  it.  The  outer 
circumference  of  the  road  is  440  yards,  and  its  width  is  60 
feet.     Find  the  area  of  the  park  exclusive  of  the  road. 

70.  A  man  agrees  to  work  for  $  3  a  day  and  his  board, 
and  to  pay  $  1  a  day  for  his  board  when  idle.  At  the  end 
of  30  days  he  receives  $  38.     How  many  days  did  he  work  ? 


Miscellaneous  Problems  243 

71.  A  ship's  chronometer  showing  Greenwich  time  was 
40  minutes  slow  Saturday  noon  and  15  minutes  fast  the 
following  Tuesday  noon.  In  what  longitude  was  the  ship 
at  each  observation  ? 

72.  A  and  B  together  erect  a  shed  at  a  cost  of  $  82.50,  A 
paying  $  22.50,  and  B  the  remainder.  They  sold  it  after- 
wards for  $  27.50.     How  much  should  each  receive  ? 

73.  At  ^1.50  per  rod,  what  is  the  difference  in  the  cost 
of  fencing  10  acres  in  the  form  of  a  square,  and  the  same 
area  as  a  rectangle  20  rods  wide  ? 

74.  A  man  travels  48  miles  due  north,  then  48  miles  due 
east,  then  28  miles  due  south.  How  far  is  he  then  from 
his  starting  point  ? 

75.  A  person  failing  in  business  owes  $  4500  and  has 
property  worth  $  1800.  How  much  will  a  creditor  receive 
whose  claim  is  $  275  ? 

76.  How  many  cubical  blocks,  each  2  inches  by  3  inches 
by  4  inches,  will  be  required  to  fill  a  space  2  feet  by  3  feet 
by  4  feet  ? 

52.   _L.  33.  31   _i_   21 

77.  Simplify  ,^1—^  X  -f—^. 

78.  Find  the  cost  of  25  boards  at  $  60  a  thousand,  each 
board  being  14  feet  long  and  14  inches  wide. 

79.  What  is  the  diagonal  of  a  rectangle  whose  sides 
measure  32  rods  and  60  rods,  respectively  ? 

80.  Find  the  capacity,  in  gallons,  of  a  rectangular  cis- 
tern 10  feet  6  inches  by  4  feet  6  inches  by  6  feet  3  inches. 

81.  A  room  24  feet  x  28  feet  and  9  feet  high  has  3  doors, 
each  3  feet  x  8  feet,  and  3  windows,  each  3  feet  x  6  feet. 
Find  the  cost  of  plastering  the  walls  and  the  ceiling  at  5  ^ 
a  square  yard,  deducting  one  half  the  area  of  the  openings. 

82.  During  a  month  of  21  school  days,  there  were  14 
girls  and  11  boys  in  attendance.     Three  girls  were  absent 


244  Arithmetic 

1  day  each,  4  boys  1^  days  each,  and  2  boys  2  days  each. 
Find  the  average  attendance  of  (a)  the  girls,  (6)  the  boys, 
(c)  the  school. 

83.  A  man  owing  $  1800  paid  his  debt  in  six  years  and 
had  $  900  left  by  saving  j\  of  his  salary.  What  was  his 
annual  salary  ? 

84.  A  certain  street  one  half  mile  long  and  6  rods  wide 
is  excavated  to  an  average  depth  of  21  feet.  Find  the  cost 
at  70  cents  a  cubic  yard. 

85.  To  make  a  certain  grade  of  concrete,  there  are  used 

2  parts  of  lime,  1  of  cement,  and  6  of  broken  stone.  How 
many  cubic  feet  of  each  are  used  in  building  a  wall  36'  x 
9'  X  li'  ? 

86.  If  milk  is  1.03  times  as  heavy  as  water,  what  is  the 
weight  of  the  milk  that  fills  a  can  28  inches  in  diameter  and 

3  feet  high  ? 

87.  A  square  prism  11  feet  high  has  a  volume  of  4851 
cubic  feet.     Find  one  side  of  the  base. 

88.  Find  the  cost  of  a  pile  of  4-foot  wood  40  feet  long  and 
6  feet  high,  at  $  5  per  cord. 

89.  A  square  field  containing  10  acres  is  divided  into 
four  equal  square  fields.  Find  the  number  of  rods  of  fence 
required  to  inclose  them. 

90.  Find  the  value  of  (0.125)^  x  (0.32)3. 

91-   Simplify  97_8l.,   ,%_> 

92.  What  will  be  the  cost  of  digging  a  ditch  30  rods  long, 
3  feet  deep,  6  feet  wide  at  the  top  and  4  feet  wide  at  the 
bottom,  at  8  cents  a  cubic  yard  ? 

93.  What  is  the  difference  between  the  length  of  a  fence 
around  a  circular   piece  of  land  comprising  43,681  square 


Miscellaneous  Problems  245 

rods,  and  the  length  of  a  fence   around  the  same  quantity 
of  land  in  the  form  of  a  square  ? 

94.  A  spherical  shell  whose  internal  diameter  is  7  inches 
is  filled  with  water.  Its  contents  are  poured  into  a  cylin- 
drical vessel  whose  internal  radius  is  7  inches.  Find  the 
depth  of  water  in  the  cylinder. 


CHAPTER   YIII. 

GENERAL   REVIEW. 
NUMBERS. 

481.  Arithmetic  is  defined  as  the  science  of  numbers;  the 
art  of  computation  by  figures. 

482.  A  number  may  be  used  to  give  the  length  of  a  line, 
the  size  of  an  angle,  the  weight  of  a  quantity  of  sugar,  the 
area  of  a  field,  etc.,  or  to  state  how  many  sheep  there  are  in 
a  flock,  people  in  a  crowd,  etc. 

483.  The  number  of  sheep  in  a  flock,  for  instance,  is  ob- 
tained by  counting ;  as  is  the  number  of  pupils  in  a  room, 
the  number  of  apples  on  a  plate,  and  the  like. 

484.  The  length  of  a  table  is  ascertained  by  compariyig  it 
witli  a  known  length  taken  as  a  standard.  A  foot  rule,  for 
instance,  is  employed,  and  if  the  length  of  the  table  contains 
the  length  of  the  ruler  exactly  four  times,  the  number  four 
expresses  the  length  of  the  table  in  terms  of  the  unit,  the 
foot. 

The  number  four  obtained  in  this  case  is  called  a  whole  number, 
because  it  contains  only  entire  units.  A  whole  number  is  also  called 
an  integer. 

485.  A  line  shorter  than  a  foot  may  be  measured  by  em- 
ploying the  divisions  of  the  foot  rule.  A  folding  rule  con- 
tains four  divisions ;  and  if  the  length  of  a  line  is  equal  to 
the  length  of  three  of  the  divisions,  the  line  is  said  to  be 
three  fourths  of  a  foot  long. 

246 


Numbers  247 


The  number  three  fourths  is  called  a  fraction,  because  it  does  not 
indicate  entire  units. 

486.  A  length  equal  to  four  times  the  foot  rule  and  three 
of  the  divisions,  is  said  to  be  four  and  three  quarters  feet 
long. 

The  number  four  and  three  quarters,  which  contains  a  whole  num- 
ber and  a  fraction,  is  called  a  mixed  number. 

487.  In  measuring  this  last  line,  the-  inch  divisions, 
marked  on  the  foot  rule,  might  be  employed  as  units  in 
determining  the  length  of  the  portion  of  the  line  remain- 
ing after  the  rule  was  employed  four  times  in  ascertaining 
the  whole  number  of  feet.  This  remaining  portion  contains 
nine  of  the  inch  divisions,  and  the  line  is  said  to  measure 
four  feet  nine  inches. 

The  expresHiou  four  feet  nine  inches  is  called  a  compound  number, 
because  it  contains  units  of  different  magnitudes,  but  of  related  kinds. 
Six  pounds  and  eight  hours  do  not  form  a  compound  number. 

488.  Things  that  can  be  measured  are  called  magnitudes 
or  quantities. 

A  magnitude  is  said  to  be  continuous  when  it  forms  a  whole  without 
presenting  any  division  ;  as,  the  area  of  a  field,  the  time  taken  by  a 
trip,  or  the  weight  of  a  body.  A  magnitude  is  said  to  be  discontinuous 
when  it  is  made  up  of  distinct  like  objects,  or  of  objects  merely  having 
the  same  name  ;  such  as,  the  steps  of  a  flight  of  stairs,  the  windows  of 
a  house,  the  cattle  in  a  drove. 

489.  A  continuous  magnitude  is  measured  by  ascertain- 
ing the  number  of  times  it  contains  a  standard  taken  as  a 
unit.  A  discontinuous  magnitude  is  measured  by  counting 
the  objects  composing  it.  Each  of  the  names  employed  in 
counting  represents  a  number. 

490.  An  abstract  number  is  one  in  which  the  kind  of  unit 
is  not  expressed;  in  a  concrete  number  the  unit  is  indicated. 


248  Arithmetic 

Thus :  five,  four  and  three  quarters,  eight,  are  abstract  numbers  ; 
eighteen  horses,  six  hours,  two  feet  eight  inches,  are  called  concrete 
numbers. 

491.  Concrete  numbers  in  which  the  units  are  those  employed  in 
measuring  continuous  magnitudes  are  called  denominate  num.bers ; 
such  as  four  acres,  twenty-five  degrees,  seven  miles. 

NOTATION  AND   NUMERATION. 

« 
Formation  of  Numbers. 

492.  It  has  been  found  possible  to  name  all  numbers  by 
the  use  of  comparatively  few  words.  One,  two,  three,  etc., 
to  ten;  hundred,  thousand,  million,  billion,  with  a  few 
others,  are  all  that  are  required  separately  or  in  combina- 
tion to  express  any  number.  Eleven  and  twelve  are  old 
forms,  meaning  ten  and  one,  ten  and  two;  the  suffix 
"teen"  means  "and  ten"  when  united  to  four,  five,  etc., 
to  nine.  The  suffix  "  ty  "  means  "  times  ten  "  when  com- 
bined with  six,  seven,  eight,  and  nine.  The  first  syllable  of 
twenty,  thirty,  forty,  fifty,  is  a  modification  of  two,  three, 
four,  and  five,  respectively. 

493.  The  nine  numbers  following  twenty  are  twenty-one, 
twenty-two,  etc.,  to  twenty-nine.  Those  following  thirty, 
forty,  etc.,  to  ninety  are  formed  in  the  same  way. 

494.  The  number  after  ninety-nine  is  one  hundred.  Num- 
bers to  one  hundred  ninety-nine  are  indicated  by  adding 
after  one  hundred  the  numbers  from  one  to  ninety-nine. 
Those  after  two  hundred,  three  hundred,  etc.,  to  nine  hun- 
dred ninety-nine  follow  the  same  rule. 

495.  After  nine  hundred  ninety-nine  comes  one  thousand. 
The  thousands  are  consecutively  numbered  to  nine  hundred 
ninety-nine  thousand,  the  next  thousand  being  called  a  mil- 
lion, which  is  equal  to  one  thousand  thousands.     One  thou- 


Notation  and   Numeration  249 

sand  millions  is  called  a  billion ;  one  thousand  billions,  a 
trillion;  etc. 

In  some  countries  a  billion  is  a  million  millions. 


Writing  Numbers. 

496.  The  necessity  of  expressing  in  writing  a  number  of 
any  size  in  the  shortest  possible  way,  led  to  the  introduc- 
tion of  characters  to  represent  a  few  important  numbers,  the 
other  numV)ers  being  formed  by  combining  these  characters. 

497.  Koman  Notation.  The  Romans  employed  seven  let- 
ters, I,  V,  X,  L,  C,  D,  and  M,  to  represent  one,  five,  ten, 
fifty,  one  hundred,  five  hundred,  and  one  thousand,  respec- 
tively. Two  was  written  II ;  and  three,  III.  Four  was 
written  IV.  Six,  seven,  and  eight  were  written  VI,  VII, 
and  VIII,  respectively,  nine  being  written  IX.  The  num- 
bers from  ten  to  nineteen  were  written  by  placing  after  X 
the  letters  representing  the  numbers  from  one  to  nine, 
inclusive.  Twenty  and  thirty  were  written  XX  and  XXX, 
forty  being  XL. 

498.  Eoman  numbers  are  written  by  expressing  the  num- 
ber of  hundreds  by  an  equivalent  number  of  C's,  substitut- 
ing, however,  a  D  for  five  C's;  by  expressing  the  tens  by 
X's,  substituting  an  L  for  five  X's ;  and  the  number  of  ones 
by  Vs,  substituting  a  V  for  five  I's.  Four  and  nine  may 
also  be  expressed  as  IV  and  IX ;  forty  and  ninety  as  XL 
and  XC ;  four  hundred  and  nine  hundred  as  CD  and  CM. 

499.  Arabic  Notation.  By  the  Arabic  system,  all  numbers 
are  expressed  by  ten  characters,  as  follows : 

0,         1,       2,        3,        4,       5,      6,        7,  8,        9 

naught    one    two    three    four    five    six    seven    eight    nine 

The  naught  (cipher  or  zero)  has  no  value  except  to  fill  a  vacant 
place  ;  the  others  are  called  significant  figures. 


250  Arithmetic 

500.  To  express  ten,  two  figures  are  used,  10.  This 
means  1  ten,  no  units ;  the  figure  in  the  second  place, 
counting  from  the  right,  represents  the  number  of  tens, 
and  that  in  the  first  place,  the  number  of  units.  The 
succeeding  tens  are  written:  20,  30,  etc.,  to  90. 

To  express  hundreds,  a  figure  in  the  third  place  is  re- 
quired; thus,  300,  500,  700,  900.  Tens  and  units  replace 
the  zeros  to  form  intermediate  numbers.  Thousands  are 
represented  by  figures  in  the  fourth,  fifth,  and  sixth  places ; 
thus,  2,000,  14,000,  256,000,  etc. 

Figures  in  the  seventh,  eighth,  and  ninth  places  represent 
millions ;  etc. 

501.  Simple  and  Local  Value.  In  the  number  60,341,  the 
1,  being  in  the  units'  place,  has  a  local,  or  place,  value  of 
one.  The  figure  4,  being  in  the  tens'  place,  has  a  local 
value  of  40.  The  figure  3,  being  in  the  hundreds'  place, 
has  a  local  value  of  300.  The  figure  6,  being  in  the  fifth 
place,  has  a  local  value  of  60,000.  The  zero  has  no  value ; 
it  merely  serves  to  show  that  there  is  no  figure  representing 
thousands  below  ten  thousand. 

502.  The  local  value  of  a  significant  figure  increases  ten- 
fold as  it  is  moved  from  right  to  left,  its  local  value  in  the 
first  place  being  the  same  as  its  simple  value. 

503.  Notation  of  Decimal  Fractions.  Our  system  of  writ- 
ing numbers  is  called  the  decimal  system,  from  the  Latin 
word  decern,  meaning  ten,  because  of  this  tenfold  increase 
in  the  local  value  of  the  figures  from  right  to  left.  Frac- 
tions made  up  of  tenths,  hundredths,  thousandths,  etc.,  of 
a  unit,  may  also  be  represented  by  figures  placed  to  the 
right  of  the  units'  place,  and  separated  therefrom  by  a 
period.     Such  fractions  are  called  decimals. 

Thus,  we  write  the  mixed  number  seventy-live  and  twenty-three 
hundredths  75.28,  the  7  representing  70,  the  5  representing  5,  the  2 
representing  2  tentlis,  the  3  representing  3  Imndredtlis. 


Notation  and   Numeration  251 

504.  The  decimal  406  ten  thousandths  is  written  .0406. 
There  being;  no  whole  number,  the  decimal  point  is  first 
written;  and  as  ten-thousandths  comprises  four  decimal 
places,  a  cipher  must  precede  the  first  significant  figure  of 
the  decimal. 

505.  Notation  of  Common  Fractions.  When  it  is  said 
that  a  certain  line  measures  five  sixths  of  a  foot,  it  means 
that  the  standard  unit,  the  foot,  was  divided  into  six  equal 
parts  called  sixths,  and  that  the  line  was  equal  in  length 
to  five  of  the  divisions. 

506.  To  write  five  sixths,  we  place  5  above  a  line  and  6 
below,  thus,  f .  A  fraction  written  in  this  form  is  called  a 
common  fraction.  The  number  below  the  line  is  called  the 
denominator ;   the  one  above  the  line  is  called  the  numerator. 

507.  In  decimals  it  is  unnecessary  to  write  the  denomina- 
tor, as  the  number  of  places  following  the  decimal  point  indi- 
cates that  the  denominator  is  1  followed  by  as  many  ciphers 
as  there  are  such  places. 

508.  In  any  fraction  the  denominator  shows  the  size 
of  the  fractional  unit;  that  is,  the  number  of  parts  into  which 
the  standard  unit  has  been  divided  ;  the  numerator  shows 
the  number  of  the  fractional  units.  The  smaller  the  denomi- 
nator, the  larger  is  the  fractional  unit;  \  being  larger  than 

509.  In  writing  a  mixed  number,  the  common  fraction 
immediately  follows  the  whole  number.  It  forms  a  portion 
of  the  unit,  not  being  counted  separately  in  determining  the 
local  value  of  the  figures  composing  the  whole  number. 

510.  A  compound  number,  such  as  3  pounds  10  ounces,  is 
written  with  no  mark  between  the  different  related  units. 
The  denominations,  however,  are  generally  abbreviated 
thus  :  3  lb.  10  oz. 


252 


Arithmetic 


511.  A  per  cent  is  written  by  placing  the  sign  %  after 
the  number.  • 

512.  Numeration  of  Integers.  Numbers  are  separated  into 
periods  of  three  figures  each,  beginning  at  the  right :  the 
first  period  representing  units ;  tlie  second,  thousands ;  the 
third,  millions ;  as  shown  below.  Each  period  is  divided 
into  units,  tens,  and  hundreds.  The  number  is  read  by 
reading  each  period  separately,  beginning  at  the  left,  and 
giving  the  name  of  each  period  as  it  is  read,  but  omitting 
the  name  of  the  units'  period. 


^^AMES  OF  Periods: 

Millions 

Thousands 
•a 

Units 

(A 

E 
0 

c 
ctf 

CO          (A 

3       TJ 

Names  of  Places  : 

-mill 
ons 

-tho 
isan 
ds 

(A 

"O      1=       ,« 

^     0     «= 

■0 

Qi       —         ^ 

Q)        2        "5 

0> 

2-         P         ^ 
T5         V         0 

>-        -C         w 

i- 

■C        r         3 

"o     <A     <2 

C         E        ■= 

E         E         0 

E         E        +i 

3         «        •— 

S         0        SI 

3         0         C 

ll     H     S 

I        h-        1- 

T      H      3 

The  Number  : 

12    3 

4    5    6 

7    8    9 

Orders : 

9th  8th  7th 

6th  5th  4th 

3d  2d  1st 

The  number  is  123  miUion,  456  thousand,  789. 

It  is  composed  of  9  units  of  the  1st  order,  8  of  the  2d,  7  of  the  3d, 
6  of  the  4th,  6  of  the  5th,  4  of  the  6th,  3  of  the  7th,  2  of  the  8th,  1  of 
the  9tli. 

513.  In  reading  a  number  containing  ciphers,  read  only 
the  significant  figures. 

308,000,020  is  read  "  three  hundred  eight  million,  twenty." 

514.  Eeading  Decimals.  A  decimal  is  expressed  in  words 
by  reading  it  as  a  whole  number  followed  by  the  denomina- 
tion of  the  last  figure,  counting  from  left  to  right,  as  shown 
on  the  next  page. 


Notation  and  Numeration  253 


GO 

a 

CO 

13 
O 


x:        a        ^       '^ 

I    1     §     i     I 

■4J  1^  -«^  -M  ,Jh 

.02046 
The  decimal  .02046  is  read  "2046  liundred-thousandths." 

515.  In  reading  a  whole  number  and  a  decimal,  the  word 
and  is  generally  used  between  the  former  and  the  latter, 
being  omitted  both  in  reading  the  whole  number  and  in 
reading  the  decimal. 

Thus,  8034.0466  is  read  "8  thousand  34  and  4  hundred  56  ten- 
thousandths." 

516.  Note.  To  avoid  confusion,  however,  it  is  advisable  to  employ 
the  word  itnits  in  such  combinations  as  the  following :  300.008,  which 
should  be  read  "300  units  and  8  thousandths."  It  may  also  be  read 
"300  and  decimal  8  thousandths." 

517.  Beading  Fractions.  A  fraction  is  read  by  reading  the 
numerator,  followed  by  the  denominator  read  as  a  fractional 
unit,  i  is  read  as  one  half;  J,  as  one  quarter;  i,  as  one 
third.  In  other  cases,  the  fractional  unit  is  expressed  by 
the  suffix  "th"  or  "ths."  A  unit  numerator  is  frequently 
read  as  "  a  " ;  31  being  called  3  and  a  half. 

Note.  The  word  and  is  employed  between  the  whole  number  and 
the  fraction  in  reading  mixed  numbers. 

518.  Eeading  Per  Cents.  Per  cents  composed  of  whole  or  , 
mixed  numbers  are  read  by  using  the  words  per  cent  after  • 
the  number. 

The  following  illustrations  will  show  the  usual  business 

method  of  reading  fractional  per  cents : 

» 

1%  is  read  "  one  half  of  one  per  cent." 
Yo%  is  read  "three  tenths  of  one  per  cent." 
.5%  is  read  "  five  tenths  of  one  per  cent." 


254  Arithmetic 

519.  Exercises,  Oral  and  Written. 

1.  Write  a  number  containing  6  units  of  the  second  order, 
4  units  of  the  third  order,  and  8  units  of  the  sixth  order. 

2.  What  number  contains  4  units  of  the  fifth  order,  7 
units  of  the  fourth  order,  and  2  units  of  the  first  order  ? 

3.  Write  14  and  77  hundred-thousandths. 

4.  In  the  number  360,912,  what  is  the  place  (local)  value 
of  the  first  two  figures,  reading  from  the  left?  What  is 
the  place  value  of  the  sixth  figure  ?  What  is  the  ratio  of 
the  place  value  of  the  second  figure  to  the  place  value 
of  the  first  figure  ?  What  is  the  place  value  of  the  first 
two  figures  as  compared  with  the  last  two?"  Of  the  second 
two  compared  with  the  last  two?  Of  the  fourth  compared 
with  the  last  two  ? 

5.  Answer  the  foregoing  questions  with  relation  to  the 
number  3609.12. 

REDUCTIONS. 

520.  Preliminary  Exercises. 

1.  How  many  units  of  the  first  order  are  there  in  3 
units  of  the  second  order  ? 

2.  In  5  units  of  the  fourth  order  ? 

3.  In  7  units  of  the  sixth  order  ? 

4.  How  many  tenths  in  1  unit  ?     In  2  units  ?  • 

5.  How  many  sixths  in  1  ?     In  2  ? 

6.  How  many  ounces  in  2  pounds  ?     In  ^  pound  ? 

7.  How  many  fourths  in  2  ?     In  2^? 

8.  How  many  inches  in  2  feet  ?     In  2  feet  3  inches  ? 

9.  How  many  hundredths  in  2  ? 

10.   What  per  cent  of  a  number  equals  3  times  the  num- 
ber ? 


Reductions  255 

521.  Keduction  to  Higher  Terms;  Reduction  Descending. 
Changing  a  given  number  of  fourths  to  an  equivalent  num- 
ber of  twelfths,  of  hundreds  to  tens,  of  tenths  to  thou- 
sandths, of  pounds  to  ounces,  of  acres  to  square  rods,  and 
the  like,  involves  in  each  case  a  change  of  unit.  In  the 
case  of  the  fraction,  it  is  called  reduction  to  higher  terms, 
because  the  terms  of  the  fraction  are  larger";  in  the  other 
instances    it   is   more  properly   called  reduction  descending. 

In  changing  3  fourths  to  twelfthi,  for  instance,  we  multiply  3 
twelfths,  the  equivalent  of  1  fourth,  by  3,  the  number  of  fourths, 
which  gives  9  twelfths  as  the  result.  To  change  2  pounds  to  ounces, 
16  ounces,  the  equivalent  of  1  pound,  is  multiplied  by  2,  the  number 
of  pounds,  which  gives  32  ounces  as  the  result. 

522.  A  whole  number  is  changed  to  a  fraction  by  multi- 
plying the  denominator  of  the  fraction  by  the  whole  num- 
ber, the  denominator  indicating  the  number  of  fractional 
units  in  the  unit  of  the  whole  number.  In  changing  a 
mixed  number  to  a  fraction  having  the  same  denominator 
as  that  of  the  fraction,  the  whole  number  is  first  reduced  to 
the  fraction,  and  the  fractional  part  of  the  mixed  number  is 
added  to  it. 

To  reduce  17f  to  a  fraction,  17  is  changed  to  fifths  by  multiplying 
5  by  17,  which  gives  85  fifths.  To  this  is  added  4  fifths,  making  89 
fifths,  written  %^-. 

In  practice,  5,  the  smaller  number  is  taken  as  the  multiplier,  and 
the  numerator,  4,  is  added  in  at  the  time  the  multiplication  is  per- 
formed. 

523.  A  fraction  in  which  the  numerator  is  equal  to  the 
denominator  or  greater  than  it,  is  called  an  improper  fraction. 
A  fraction  Avhose  numerator  is  less  than  the  denominator  is 
called  a  proper  fraction. 

524.  Oral  Exercises. 

1.  Reduce  22|  to  an  improper  fraction. 

2.  Change  -^J-  to  eighths. 


256  Arithmetic 

3.  How  many  ounces  in  10  lb.  13  oz.  ? 

4.  Change  10  acres  to  square  rods. 

5.  How  many  tens  in  316  thousands  ? 

6.  How  many  units  of  the  second  order  are  equivalent 
to  8  units  of  the  fifth  oider  and  3  units  of  the  third  order  ? 

7.  Read  3  million  as  tens.    As  hundreds.    As  thousands. 
As  tenths. 

8.  Change  J  to  hundre^dths. 

9.  Express  |^  as  a  per  cent. 

10.  Express  i  as  a  decimal. 

11.  Reduce  .125  lb.  to  ounces. 

12.  How  many  sq.  rd.  in  J  acre  ? 

13.  Change    |   to   sixths.      To    ninths.      To   12ths.     To 
24ths.     To30ths.     To.  120ths. 

14.  How  many  thirds  in  31? 

15.  Change  42  to  an  improper  fraction  having  10  for  a 
denominator. 

Eeduction  to  Lower  Terms ;  Reduction  Ascending. 

525.   Preliminary  Exercises. 

1.  Reduce  316,000  to  hundreds. 

2.  Change  1600  sq.  rd.  to  acres. 

3.  Reduce  173  ounces  to  pounds  and  ounces. 

4.  Reduce  ^-^-  to  an  improper  fraction  in  lower  terms. 

5.  Change  ^-^  to  a  mixed  number. 

6.  How  many  units  of  the  fifth  order  are  equal  to  800 
units  of  the  third  order  ? 

7.  Change  -^^  to  a  wh'ole  number. 

8.  Reduce  14  ounces  to  the   fraction  of  a  pound.     To 
the  decimal  of  a  pound.     To  the  per  cent  of  a  pound. 


Reductions 


257 


9.   Keduce   ff^  *o   ^^^^s.     To  30ths.      To  loths.     To 
lowest  terms. 

10.    What  fraction  of  a  day  is  4  hr.  48  min.  ? 

526.  A  fraction  is  said  to  be  expressed  in  lowest  terms 
when  the  numerator  and  the  denominator  are  the  smallest 
possible  whole  numbers,  the  unit  of  the  denominator  in 
this  case  being  the  largest  one  possible.  When  the  denomi- 
nation of  a  compound  number  is  changed  from  a  smaller 
unit  to  a  larger  one,  the  process  is  called  reduction  ascend- 
ing. 

527.  To  change  a  fraction  to  lower  terms,  the  numerator 
and  the  denominator  are  divided  by  the  same  numbers. 

Thus,  -f^^^  =  17^  _  |5  _  7^  tiig  change  in  each  case  being  effected 
by  dividing  both  terms  by  5.  When  the  denominator  is  divided  by 
5,  the  fractional  unit  is  made  5  times  as  large  ;  one  fifth  of  the  num- 
ber of  units  in  the  numerator  must  be  taken  to  keep  the  fraction  equal 
to  the  original.  875  thousandths  =  175  two-hundredths  =  35  fortieths 
=  7  eighths.  In  the  same  way,  875  mills  =  87  cents  6  mills  =  8  dimes 
7  cents  5  mills.  In  this  case,  however,  there  are  three  different  de- 
nominations, the  result  being  a  compound  number. 

528.  Oral  Exercises. 

1.  Change  4900  sq.  rd.  to  acres,  etc. 

2.  Reduce  |f^  to  lowest  terms. 

3.  Change  44  hours  to  the  fraction  of  a  day. 

4.  Reduce  46  days  to  weeks  and  days. 

5.  What  %  of  a  day  is  1\  hours  ? 

6.  Change  1  ft.  6  iu.  to  the  fraction  of  a  yard. 

7.  What  decimal  of  a  pound  is  17s.  6(Z.  ? 

8.  Change  \t  to  lowest  terms. 

9.  What  fraction  of  a  cubic  yard  is  18  cubic  feet? 
10.  What  decimal  of  a  mile  is  40  rods  ? 


258  Arithmetic 

FACTORS   AND   MULTIPLES 

529.  The  factors  of  a  number  are  those  numbers  of  which 
the  given  number  is  the  product. 

Every  number  is,  of  course,  divisible  by  itself  and  1  ;  these  will, 
therefore,  be  omitted  from  consideration  in  determining  the  factors  of 
a  number. 

Prehminary  Exercises. 

530.  Find  the  factors  ^of  the  following  numbers  : 

1.  14  5.  35  9.  51  13.  74 

2.  21  .  6.  39  10.  57  14.  77 

3.  25  7.  46  11.  60  15.  82 

4.  33  8.  49  12.  69  16.  85 

531.  The  following  numbers  have  three  factors  each; 
what  are  they  ? 

1.  8  ^  5.  66  9.  99  13.  50 

2.  12  6.  78  10.  63  14.  75 

3.  30  7.  20  11.  28  15.  70 

4.  42  8.  45  12.  44  16.  98 

532.  A  prime  number  is  a  number  that  cannot  be  separated 
into  integral  factors  each  greater  than  1 ;  such. as  1,  2,  3,  5, 
7,  etc.  Numbers  that  are  not  prime  are  called  composite 
numbers ;  such  as  4,  6,  8,  9,  10,  etc. 

533.  Divisibility  of  Numbers.  —  A  number  is  divisible  by  2 
if  it  ends  in  0,  2,  4,  6,  8.  It  is  divisible  by  5  if  it  ends  in  0 
or  5.  A  number  is  divisible  by  3  or  9  if  the  sum  of  its 
digits  (figures)  is  divisible  by  3  or  9. 

A  number  is  divisible  by  4  if  its  last  two  figures  are  divisible  by  4  ; 
by  8,  if  its  last  three  figures  are  divisible  by  8.  An  even  number  is 
divisible  by  0  if  the  sum  of  its  digitsris  divisible  by  3 ;  etc. 


Factors  and    Multiples  259 

534.    Oral   Exercises. 

Which  of  tlie  following  numbers  is   divisible  by    2,   by 
3,  by  5,  by  9? 


1.  120 

5.  1825- 

9. 

11,250 

2.  475 

6.  4684 

10. 

25,065 

3.  570 

7.  2346 

11. 

33,333 

4.  243 

8.  1234 

12. 

25,942 

535.  To  find  the  prime  factors  of  a  number,  divide  it 
by  any  prime  number ;  divide  the  quotient  in  the  same 
manner ;  continue  to  divide  until  a  quotient  is  obtained 
that  is  a  prime  number.  The  divisors  and  the  last  quotient 
will  be  the  prime  factors. 

536.  Written  Exercises. 

Find  the  prime  factors  of  the  following  : 

1.  630.     Ans.     2x3x3x5x7. 

2.  798      4.  1000      6.  3672      8.  1836 

3.  350      5.  1750      7.  5000      9.  1650 

537.  Oral  Exercises. 

What  numbers  will  divide  36  ? 

Since  36  =  2  x  2  x  3  x  3,  any  number  that  can  be  formed  from 
these  factors  will  be  a  divisor.  They  are  2,  3,  2  x  2,  2  x  3,  3  x  3, 
2  X  2  X  3,  2  X  3  X  3.     Ans.  2,  3,  4,  6,  9,  12,  and  18. 

Find  the  divisors  of  : 

1.  24  3.    48  5.    60  7.    40 

2.  50  4.    32  6.    45  8.    70 

538.  Give  the  numbers  that  will  divide  both: 
The  smaller  number  may  be  given  as  one  divisor. 

1.  24  and  36  4.    42  and  84  7.    21  and  63 

2.  18  and  27  5.    16  and  56  8.    36  and  56 

3.  32  and  60  6.    15  and  75  9.    27  and  81 


54 

168. 
2 

to.    Find  the 
84  -  126  -  X^^ 

3 

42-    63 

7 

14-    21 

2  xc 

2-      3 
;  X  7  =  42.    Ans 

260  Arithmetic 

539.  The  greatest  common  divisor  of  two  or  more  numbers 
is  the  greatest  number  that  will  exactly  divide  each. 

Find  the  greatest  common  divisor  of  84,  126,  and 

Place  the  numbers  in  a  row,  and  cancel  168. 
Since  tiiis  is  a  multiple  of  84,  it  will  be  divis- 
ible by  any  number  that  will  divide  84. 

Divide  84  and  126  by  any  prime  number  that 
is  exactly  contained  in  both,  say  2.  Divide 
the  quotients  in  the  same  way,  say  by  3. 
Divide  the  new  quotients  by  7,  the  only  prime  number  that  is  a 
common  factor  to  both.  The  greatest  common  divisor  is  2  x  3  x  7, 
or  42,  the  product  of  the  three  common  prime  factors, 

541.  The  chief  use  of  the  greatest  common  divisor  in 
arithmetic  is  in  the  reduction  of  fractions  to  lowest  terms. 
This,  however,  can  generally  be  done  more  quickly  by  divid- 
ing both  terms  by  a  common  factor,  continuing  the  process 
until  the  fraction  can  no  longer  be  reduced. 

542.  Written  Exercises. 
Reduce  to  lowest  terms  : 

4     12.0  7      _6_0 

^'     26¥  '•     133" 

K      4  84  Q      108 

O.     6  72  °*     144 

fil08  q288 

543.  Another  method  of  finding  the  greatest  common 
divisor  of  two  numbers  is  by  dividing  the  greater  by  the 
less,  and  the  first  divisor  by  the  first  remainder,  the  second 
divisor  by  the  second  remainder,  and  so  on,  until  there  is  no 
longer  a  remainder,  in  which  case  the  last  divisor  is  the 
greatest  common  divisor. 

This  method  is  shown  in  the  next  example,  in  which  |f  jf  is  reduced 
to  lowest  terms  by  first  finding  the  greatest  common  divisor  of  2916 
and  3072, 


1. 

600 
1320 

2. 

432 
TT0¥ 

3. 

54  0 
TT67 

Factors  and   Multiples  26 


544.    Reduce  f  |ff  to  lowest  terms : 


1 

2916)3072 

2916   18 

*   156)2916 

156 

1356 

1248 

1 

108) 

156 

108 

2 

48) 

108 

96 

J 

Dividing  3072  by  2916  gives  a  re- 
mainder of  156,  the  new  divisor. 
This  is  contained  in  2916,  18  times 
with  a  remainder  of  108,  the  next 
divisor.  The  next  remainder,  48, 
becomes  the  divisor,  leaving  a  re- 
mainder of  12.  This,  being  con- 
tained exactly  in  48,  is  the  greatest 
common  divisor  of  2916  and  3072. 


12)48 

48 


Dividing  both  terms  of  the  fraction  by  12,       oq-xr  .  io      oiq 
we  get  the  egjuivalent  fraction  in  lowest       qn79  •  19  ~  9^'    '^"^' 
terms. 


545.  Reduce  to  lowest  terras  : 

1   3  5  3_5  O   2047  c   164  9 

^'     8989  "*•  2231  ^-  T9F9 

9   793  A       153  ft   3  2  3 

'^•1950  *•  49  3  "•  219"^ 

546.  A  multiple  of  a  number  is  produced  by  multiplying 
the  given  number  by  an  integer.  .12  is  a  multiple  of  4,  of 
6,  of  2,  of  3.  It  is  a  common  multiple  of  all  of  these  num- 
bers ;  24,  36,  48  are  common  multiples  of  the  foregoing 
numbers.  The  smallest  number  that  will  contain  each  is  12, 
which  is  called  the  least  common  multiple. 

547.  Preliminary  Exercises.  ^ 

1.  Give  three  numbers  that  will  contain  8  and  6. 

2.  What  is  the  smallest  number  that  will  contain  8  and  6  ? 

3.  Find  the  least  common  multiple  of  6,  8,  and  12. 

4.  Give  two  common  multiples  of  10,  20,  and  30. 

5.  What  is  the  least  common  multiple  of  10,  20,  and  30  ? 


262  Arithmetic 

548.  Find  the  least  common  multiple  of  6,  S,  12,  16,  and 
20. 

^  =  Writing  tlie  numbers  in  a  column,  find 

^  =  the    prime    factors    of    each,    cai^eling 

12  =  2x2x3  =  22  x3      6    and    8,    which    are   contained   in    12 

16  =2x2x2x2=  2^      and  16,  respectively.    The  least  common 

20  =  2  X  2  X  5  =  22  X  5      multiple  of  the  remaining  numbers  must 

contain  2*,  3,  and  5  ;  it  is  therefore  a  product  of  these  factors. 

24  X  3  X  5  =  210.    Ans. 

549.  Find  the  least  common  multiple  of  12,  24,  36,  45, 
and  60. 

2)1^  -  24  -  36  -  45  -  60 

2)12  -  18  -  45  -  30 

3)6-    ^_45_;^ 

2  -15 

2x2x3x2x15  =  360.    Ans. 

Another  method  is  to  divide  the  numbers  by  any  prime  number 
that  is  a  factor  of  at  least  two  of  the  numbers,  first  canceling  any  of 
the  numbers  that  i§  contained  in  any  other ;  as,  for  instance,  12  in  this 
example.  Taking  2  as  a  divisor,  the  quotients  are  written  underneath, 
45  being  brought  down,  since  it  is  not  a  multiple  of  2.  2  is  again  used 
as  a  divisor,  giving  quotients  as  shown,  45  being  again  brought  down. 
9  and  15  are  canceled,  being  factors  of  45,  and  3  is  taken  as  the  next 
divisor.  There  being  no  common  factor  of  the  remaining  numbers,  2 
and  15,  the  least  common  multiple  is  obtained  by  multiplying  together 
the  three  divisors  and  the  two  final  quotients. 

550.  Written  Exercises. 

Find  the  least  common  multiple: 

1.  Of  9,  15,  18,  and  27. 

2.  Of  2,  3,  4,  5,  6,  and  7. 

3.  Of  125,  200,  250,  and  300. 

4.  Of  240,  324,  and  120. 

5.  Of  45,  60,  72,  18,  and  12. 


Addition  263 

551.  Miscellaneous  Examples. 

1.  Give  the  prime  factors  of  3696. 

2.  Find  three  equal  factors  of  729. 

3.  What  is  the  5th  power  of  four  ? 

4.  Find   the   greatest   common    divisor    of    8947    and 
10,603. 

5.  Find  the  least  common  multiple  of  38  and  57. 

6.  What  are  the  two  equal  factors  of  2304  ? 

7.  Extract  the  square  root  of  4096. 

8.  What  is  the  cube  root  of  19,683  ? 

9.  Find  the  5th  root  of  3125. 
10.    Is  1537  a  prime  number  ? 

THE  FUNDAMENTAL   PROCESSES. 
ADDITION. 

552.  The  process  of  finding  a  number  whose  value  is  the 
combined  values  of  two  or  more  given  like  numbers  is  called 
addition.  The  numbers  to  be  added  are  called  addends. 
The  number  obtained  by  addition  is  called  the  sum,  or 
amount.  The  sign  of  addition  is  +,  called  plus,  which  is 
a  Latin  word  meaning  "  more." 

553.  The  addends  must  be  like  numbers ;  that  is,  they 
must  all  be  concrete  numbers  having  the  same  unit,  or  ab- 
stract numbers.  Concrete  numbers  of  different  related  units 
must  be  reduced  to  the  same  unit  before  they  can  be 
added. 

Note.  A  given  number  of  boys  and  a  given  number  of  girls  may 
be  added  to  ascertain  the  number  of  pupils  in  a  class  ;  but  they  are 
considered  as  having  the  same  unit,  viz.,  pupils. 


264  Arithmetic 

554.  Addition  of  Integers. 

Add  376,  1835,  29,  647,  and  3324. 

376       31  units  The  numbers  are  written  in  a  row,  the  units, 

1835     ^^  ^^"^  tens,  hundreds,   and  thousands   of  each   addend 

^^     20     hundreds        ,....,  ,  , .      ,  r.,, 

^^    4      thousands    Standing  m  the  same  column,  respectively.     The 

647    6211  sum  of  the  units'  column  is  31,  which  is  equal  to 

3324  3  tens  1  unit.     Placing  1  in  the  column  of  units, 

6211  3  is  added  to  the  column  of  tens,  making  a  total 

of  21  tens,  equal  to  2  hundreds  1  ten.  Placing  1  in  the  column  of 
tens,  2  is  added  to  the  column  of  hundreds,  making  a  total  of  22  hun- 
dreds, equal  to  2  thousands  2  hundreds.  Placing  2  in  the  column  of 
hundreds,  2  is  added  to  the  column  of  thousands,  making  6  thousands. 
Placing  6  in  the  column  of  thousands  gives  the  result  as  6211. 

The  small  numbers  written  to  the  right  show  the  totals  of  each 
column.     Their  sum  gives  the  total  of  all  the  numbers. 

555.  In   adding  long  columns,  it  is  the  practice  of  ac-     376   31 

countants    to    place   alongside    the   sum    of    each   column,    1835    ^^ 

22 
including  the  number    "carried."     This  enables  them,  in       29     . 

the  case  of  an  error  in  the  total  of  one  column,  to  obtain     647 

the  correct  result  without  going  over  all   of   the  previous   3324 

columns  to  ascertain  the  "carrying"  figures.  6211 

556.  The  pupils  should  employ  as  few  words  as  possible  in  adding. 
In  the  example  given,  it  is  necessary  in  adding  aloud,  at  the  black- 
board, for  instance,  to  say  only,  11,  20,  25,  31 ;  5,  9,  11,  14,  21  ;  5, 
11,   19,  22;  5,  6. 

557.  To  prove  the  correctness  of  the  result  in  addition, 
the  columns  may  be  added  doivn  the  second  time,  if  the  first 
result  is  obtained  by  adding  ^ip.  Another  plan  is  to  divide 
a  very  long  column  into  two  portions,  add  each  separately, 
and  combine  the  sums.  If  the  answer  obtained  in  this  way 
agrees  with  the  first  answer,  the  work  may  be  presumed 
to  be  correct. 

558.  In  adding  two  large  numbers  mentally,  a  pupil  should  add  to 
the  first  number  the  hundreds  of  the  second,  then  the  tens,  and  lastly 
tlie  units.  Thus,  the  sum  of  375  ivud  256  is  most  readily  found  by 
thinking,  375  +  200  +  50  -f-  6, 


559.   Oral  Exercises. 

1.  394  +  200 

4.  583  +  300 

7.  642  +  200 

10.  489  +  400 


Addition 

2.   394  +  240 

5.   583  +  350 

8.    642  +  270 

11.' 489  +  440 


265 

3.  394  +  247 

6.  583  +  358 

9.  642  +  279 

12.  489  +  445 


348. 

25.6 
4.328 
506.27 

14. 
898.198 


560.  Addition  of  Decimals.  Decimals  are 
added  in  the  same  way  as  integers,  care  being 
taken  to  have  like  units  in  the  same  column. 
The  decimal  point  acts  as  a  guide,  a  point  being 
placed  after  each  integral  addend. 


561.    Addition  of  Mixed  Numbers. 
Add  17f ,  14A  9|,  and  6i. 

17|  =  ll^Yu  "^o  ^^^  t^^  s^"^  of  ^^6  fractions,  they  are 

144  —  1496- 


9|  = 


I   80l 
'120 


first  reduced  to  fractions  having  a  common 
denominator. 


"2  —      "T20 

40  4 1 


Ans. 


120ths 

m 

45 

iH 

96 

n 

80 

6i 

60 

48rViT 

2  8  1    —  0  4  1 
T'2  0   —  "^T2  0 

In  this  example  120  is  the  least  common 
denominator.  The  sum  of  the  fractions  is 
f  |i,  which  is  equal  to  2xV(j.  The  fraction 
is  written  in  its  place,  and  2  is  carried  to 
the  units'  column  of  the  whole  numbers. 

The  arrangement  at  the  right  facilitates       Ans. 
the  addition  of  the  numerators  by  writing 
the  common  denominator  above  the  numerators. 

Note.  The  least  common  multiple  of  the  denominators  is  the  least 
common  denominator  of  the  fractions.  This  should  be  determined  by 
inspection,  if  possible. 

562.    Written  Exercises. 
1. 


2.  12|  +  7|  +  95i  +  3f 

3.  42^  +  15^fo+623-V¥o  +  59 

4.  321y\  +  105/oV  +  64| 


10  oz. 

5  pwt. 

17  gr. 

3 

18 
12 

20 

5 

3 

266  Arithmetic 

563.  Addition  of  Compound  Numbers. 

Add  the  following : 

Units  of  each  denomination  are  placed 
in  separate  columns.    Those  of  the  lowest 
denomination  are  then  added,  and  if  the 
sum  exceeds  the  number  of  units  in  the 
18  oz.     19  pwt.     13  gr.       next  higher  denomination,  it  is  reduced 

to  the  units  of  this  denomination,  and 
the  remainder  is  written  under  that  column.  The  number  of  units 
of  the  higher  denomination  obtained  by  the  reduction  is  "carried"  to 
the  next  column,  etc. 

564.  "Written  Exercises. 

1.  14  sq.  yd.  20  sq.  ft.  100  sq.  in.,  15  sq.  ft.  81  sq.  in., 

10  sq.  yd.  18  "^sq.  ft. 

2.  15  A.  108  sq.  rd.,  24  A.  20  sq.  yd.,  15  A.  80  sq.  rd. 
15  sq.  yd. 

3.  17  bu.  3  pk.  2  qt.,  58  bu.  7  qt.,  35  bu.  2  pk.  7  qt., 

11  bu.  4  qt. 

4.  27  rd.  3  yd.,  119  rd.  2  ft.,  277  rd.  4  yd.  1  ft.,  160  rd. 

5.  46  gal.  2  qt.  1  pt.,  33  gal.,  23  gal.  3  qt.  1  pt.,  96  gal. 
1  qt. 

SUBTRACTION. 

565.  Considering  subtraction  as  the  reverse  of  addition,  it 
may  be  defined  as  an  operation  to  find  the  remaining  quantity, 
when  the  sum  of  two  like  quantities  is  given,  and  one  of 
the  quantities. 

566.  Subtraction  of  Integers  and  Decimals. 

From  1246.13  take  987.542. 

Minuend  1240  13  ^^    ^^    addition,   like    units    are 

Subtrahend  987  542  placed  in  the  same  column.     Assum- 

Remainder,'         258*588    Ans.     "^^  ^^^^  ^^^  minuend  is  the  sum  of 

the  subtrahend  and  the  remainder,  a 
number  is  to  be  found  that,  added  to  987.542,  will  give  1240.13.     The 


Subtraction  267 

result  may  be  obtained  by  placing  under  each  figure  of  the  subtrahend 
the  figure  which,  combined  with  it,  will  produce  the  corresponding 
figure  of  the  minuend,  care  being  taken  to  provide  for  the  carrying 
figures.  Beginning  at  the  left,  say  2  and  8  (writing  8)  are  10  ;  carry- 
ing 1,  5  and  8  (writing  8)  are  13  ;  carrying  1,  6  and  5  (writing  5)  are 
11 ;  carrying  1,  8  and  8  (writing  8)  are  16  ;  carrying  1,  9  and  5  (writ- 
ing 5)  are  14;  carrying  1,  10  and  2  (writing  2)  are  12. 

While  this  method  is  more  easily  applied  by  young  pupils,  it  is  less 
frequently  used  in  this  country  than  the  one  next  given. 

567.  Considering  subtraction  as  taking  one  number  from  another, 

*  the  process  may  assume  this  form:    Since 

111815   10  12  ^  ,  ,       -^  ,     ^     ,  ,  ,        , 

1  3  5    0    2 10  ^  thousandths   exceeds  0   thousandths,  the 

X  2  4  0  .  ^  3  3   hundredths  is  reduced  to  2  hundredths 

9  8  7.542  ^iid  10  thousandths  ;  2  thousandths  from  10 

2  5  8.588     Ans.      thousandths  leaves  8   thousandths,   written 

in  the  thousandths'  column  of  the  result. 
Since  4  hundredths  exceeds  2  hundredths,  1  tenth  is  reduced  to  10 
hundredths  and  added  to  the  2  hundredths,  making  12  hundredths  and 
leaving  0  tenths  in  the  minuend  ;  4  hundredths  from  12  hundredths 
leaves  8  hundredths.  Since  o  tenths  exceeds  0  tenths,  the  6  units  is 
reduced  to  5  units  and  10  tenths,  etc. 

In  this  example  the  line  of  small  figures  next  above  the  minuend 
shows  the  successive  reductions,  the  three  in  the  hundredths'  column 
being  canceled  and  replaced  by  2,  and  10  thousandths  being  written 
in  the  proper  column.  When  the  2  hundredths  is  increased  to  12 
hundredths,  the  1  tenth  is  canceled  and  0  written  above  it.  This  is 
next  changed  to  10  tenths,  the  6  units  being  changed  to  5,  and  so  on. 

Note.  It  is  understood,  of  course,  that  the  small  figures  illustrat- 
ing the  process  should  not  be  employed  in  doing  the  work. 

568.  Still  another  method  is  employed  :  a  figure  of  the  minuend 
r>  14  16    11 13  10  ^^  increased  by  10  units  of  the  next  order 

1  2  4  ^  .  ;[  3  when  necessary,  and  the  next  .order  in^he 

10  9  8     6  5  subtrahend  is  increased  by  1.    This  method, 

— "  P  '^  •  ^  4  w  though  more  difficult  to  explain,  is  just  as 

2  5  8.088     Ans.         logical  as  the  other,  and  is  somewhat  easier 

in  practice.     It  renders  unnecessary  a  change  in  any  figure  of  the 

minuend  other  than  to  prefix  1  to  it  when  necessary,  in  which  case 

the  next  figure  of  the  subtrahend  is  increased  by  1. 


268  Arithmetic 

569.  Oral  Exercises. 

1.  641-300-80-7  5.  742-387 

2.  932-400-60-5  6.  824-465 

3.  820-500-70-3  7.  940-573 

4.  764-200-90-5  8.  653-295 

570.  Subtraction  of  Mixed  Numbers. 
From  2681  take  199if . 

24ths         Make     the    fractions    similar  by   reducing 


2681 


Ans.     68i 


q  them  to  a  common  denominator.     Since  3  is 

iq  less  than  19,  the  8  units  in  the  minuend  is  re- 

_^_  _  1  duced  to  7||.  This  added  to  the  21  makes 
^*      ^     7f^.     From  f|  taking  \\  leaves   2^,   which  is 

written  in  the  result,  after  being  reduced  to  |.     267  —  199  gives  the 

integral  part,  68. 

571.  Subtraction  of  Compound  Numbers. 

From     10  oz.     7  pwt.   10  gr.  Si'^ce  10  gr.    is  less  than  20  gr.,   it 

Take       5  oz.   15  pwt.  20  gr.      must  be  increased  by  1  pwt.,  or  24  gr., 

Ans.    4  oz.  11  pwt.  14  gr.      leaking  34  gr.     34  gr.  -  20  gr.  =  14  gr., 

written  in  the  result.  The  6  pwt.  re- 
maining is  increased  by  1  oz.,  or  20  pwt.,  making  26  pwt.  26  pwt. 
—  15  pwt.  =  11  pwt.,  written  in  the  result.  5  oz.  from  the  remaining 
9  oz.  leaves  4  oz.,  written  in  the  result. 

572.  "Written  Exercises. 


1. 

1 84-3 28-7- 

6. 

16  rd.  3  yd.-4rd.  2  ft. 

2. 

16 -.00462 

7. 

(7.5-3) -(.8 -.09) 

3. 

OCA   3     _  1 4fi     2  7 

8. 

21i-(16i-3yV) 

4. 

1  sq.  yd.  — 15  sq.  ft.  94 

sq. 

in. 

9. 

(4i-lf)-(20i-18f) 

5. 

3i-  A  —  475  sq.  rd. 
573.    Oral  Problems. 

10. 

1  -  (.032  -  .0075) 

1.  The  temperature  yesterday  was  18°  above  zero;  to-day 
it  is  3°  below  zero.  How  many  degrees  is  to-day  colder 
than  yesterday  ? 


Multiplication  269 

2.  Find  the  difference  in  latitude  between  a  place  in  G0° 
north,  and  another  in  35°  south  latitude. 

3.  If  A  is  worth  $  75  and  B  has  nothing  and  also  is  in 
debt  $30,  how  much  poorer  is  B  than  A  ? 

4.  M  travels  east  20  miles  on  a  road,  then  25  miles  west 
on  the  same  road,  then  5  miles  west,  then  45  miles  east ;  how 
many  miles  is  he  from  his  starting  point  ?  How  far  has  he 
traveled  ? 

MULTIPLICATION. 

574.  Multiplication  is  the  process  of  obtaining  a  number 
called  the  product  by  performing,  on  a  number  called  the 
multiplicand,  the  operation  that  must  be  performed  upon 
unity  to  produce  the  multiplier. 

575.  Multiphcation  of  Integers  and  Mixed  Numbers.  When 
the  multiplier  is  an  integer,  the  product  is  equal  to  the  sum 
of  the  multiplicand  used  as  an  addend  as  many  times  as  there 
are  units  in  the  multiplier.  As  the  process  of  obtaining  the 
product  by  addition  is  a  long  one,  the  following  method  is 
used. 

1.  Multiply  2468  by  3078. 

The  product  of  2468  by  8  units  is  first  obtained. 
The  product  by  7  tens  being  tens,  the  first  figure 
of  this  product  is  placed  in  the  tens'  column.  The 
first  figure  of  the  product  by  3  tliousands  is  placed 
in  the  thousands'  column.  The  sum  of  the  partial 
7596504  Ans.  products  is  the  product  required. 

2.  Multiply  567J  by  27. 

567f 

27 

27  times  3  fourths  is  81  fourths,  or  20^,  which  is  the 


2468 

X  3078 

19744  units 

17276 

tens 

7404 

thousands 

■^ .       first  partial  product.     The  product  by  7  units  is  next 

4  ^^^1  s    written,  the  first  figure  being  placed  also  in  the  column 
3969    units    ^^  ^^^^-^^ 

1134      tens 

15329^  Ans. 


4251 

product  by  | 

3969 

product  by  7 

1134 

product  by  20 

270  Arithmetic 

3.    Multiply  567  by  27f. 
567 

27|  Since  567  times  3  fourths  has  the  same 

4)1701    product  by  3       number  of  fourths  as  3  times  567,  this 

product  is  first  written,  and  is  then  re- 
duced to  a  mixed  number  by  dividing 
by  4. 
Ans.    15734^  product  by  27| 

Note.    When  both  the  multiplier  and  the  multiplicand  are  mixed 
numbers,  their  product  is  obtained  by  the  method  given  below. 

576.   Multiplication  of  Tractions. 

1.  Multiply  A  by  |. 

^28  7  multiplied  by  f  means  -|  of  f.     As  1  seventh 

^  ^  r  =57*  -^^^-  means  one  of  the  seven  equal  parts  into  which  a 
unit  is  divided,  1  fifth  of  1  seventh  means  that 
this  seventh  is  divided  again  into  5  parts,  each  of  which  is  1  thirty- 
fifth  of  the  unit.  Therefore  1  fifth  of  4  sevenths  is  4  thirty-fifths  and 
2  fifths  of  4  sevenths  is  2  times  4  thirty-fifths,  which  is  8  thirty-fifths. 

2.  Multiply  I  by  f., 

2  To  multiply  f  by  |  is  to  find  f  of  |.     1  fourth 

®  X  -  =  -    Ans.      of  8  ninths  is  2  ninths,  and  3  fourths  of  8  ninths  is 


3 


3  times  2  ninths,  or  2  thirds. 


Multiply  12|  by  51 

12f  Sometimes  one  mixed  number  may  be  multi- 

6^  plied  by  another  in  the   manner  shown  in  the 


41  =  12|  X    I    accompanying  example;  as  a  rule,  however,  the 
63|  =  12|  X  5      mixed  numbers  should  be  reduced  to  improper 
Ans.  68    =  12f  x  5^   fractions, 

17      4 
12|x  51  =^x^  =  68.     Ans. 

577.  To  multiply  one  fraction  by  another,  place  the  prod- 
uct of  the  numerators  over  the  product  of  the  denomi- 
nators, canceling  when  possible. 


Multiplication  271 

578.    Multiplication  of  Decimals. 

Multiply  .246  by  .307. 
.246  Written  as  common  fractions,  the  process 


.807 

1722  millionths 
738      ten-thousandths 

is  as  follows  : 

246  ^  307 
1000      1000 

ousandths  1000^1000      1,000,000* 

.075522    Alts.  rpj^^  multiplicand  and  the  multiplier  each 

having  three  decimal  places,  the  denominator  is  1  with  three  ciphers 
in  each  case,  and  the  product  of  the  denominators  will  be  1  with  three 
ciphers  +  three  ciphers.  To  write  this  as  a  decimal  requires  six 
decimal  places,  or  one  place  more  than  the  numerator  contains ;  the 
missing  place  is  supplied  by  a  decimal  cipher. 

579.  To  multiply  one  decimal  by  another,  proceed  as  in 
the  multiplication  of  whole  numbers,  and  point  off  in  the 
product  as  many  decimal  places  as  there  are.  in  the  multi- 
plier and  the  multiplicand  together. 

580.  rinding  a  Per  Cent  of  a  Number. 

What  is  7%  of  $347? 

^  '^^^  To  find  7  %  of  S  .347,  multiply  the  latter  by  .07,  7  %  mean- 

ing 7  hundredths,  or  .07. 


$24.29 

581.    Multiplication  of  Compound  Numbers. 

Multiply  £4  16s.  Sd.  by  7. 

7  times  8d.  =  56fZ.  =  4s.    8d.     Write  Sd.  and 

£4  16s.  8d.  carry  4s.     7  times  16s.  =  112s.    Adding  4s.  gives 

X  7  116s.  =£5   16s.     Write  16s.  and  carry  £5.     7 


£33  16s.  8d.    Ans.     times  £4  =  £28.     Adding  £5  gives  £33,  which 
is  written  in  the  result. 

582.  The  multiplicand  may  be  concrete  or  abstract;  the 
multiplier  is  always  abstract. 

When  the  multiplicand  is  concrete,  the  product  is  a  like 
concrete  number. 

While  we  say,  for  instance,  that  the  cost  of  1874  articles  at  $5  each 
is  1874  times  •'$5,  in  practice  we  employ  5  as  the  multiplier. 


272  Arithmetic 

583.  The  multiplication  sign,  x ,  is  generally  read  "  mul- 
tiplied by,"  7  lb.  X  5  =  35  lb.,  meaning  7  pounds  multiplied 
by  5  equals  35  pounds.  As  the  product  of  two  numbers  is 
the  same,  no  matter  what  is  the  order  of  the  factors,  the 
foregoing  is  sometimes  written  5  x  7  lb.  =  35  lb.,  in  which 
case  the  sign  must  be  read  "times";  that  is,  5  times  7  lb.  = 
35  lb. 

Note.  In  expressions  such  as  8  ft.  x  7  ft.,  the  sign  x  should  be 
read  "  by  "  ;  thus,  3  ft.  by  7  ft. 

584.  Short  Methods  in  Multiplication. 

Pupils  should  have  occasional  drills  in  multiplying  numbers  without 
unnecessarily  using  their  pencils.     A  few  types  are  here  given. 

Sight  Exercises. 


1. 

24  X  .25 

5. 

36x25 

9. 

64  X  .375 

2. 

72  X  .125 

6. 

48  X  125 

10. 

64  X  3.75 

3. 

331%  of  66 

7. 

69  X  33i 

11. 

64  X  37.5 

4. 

12i%  of  88 

8. 

72  X  121 

12. 

371  X  64 

48  X  Hi 

19.   16x611 

66  X  32J 

20.    16x861 

16  X  36i 

21.   88x991 

A  number  is  multiplied  by  99  by  annexing  two  ciphers  to  it  and 
deducting  the  original  number ;  that  is,  99  times  96  equals  9000  —  96. 

88  X  49  =  (1  of  8800)  -  88.  48  x  111  =  (^  of  48OO)  -  48.  27  x  32i 
=  (1  of  2700) -27. 

13.  99  X  99  16. 

14.  88x49  17. 

15.  36  X  24  18. 

The  product  of  two  numbers  whose  tens'  figure  is  the  same  and  the 
sum  of  whose  units'  figures  is  10,  may  be  obtained  by  prefixing  to  the 
product  of  the  units'  figures  the  ijroduct  of  the  tens'  figure  by  itself 
increased  by  1.  Thus,  84  times  86  may  be  written  directly  by  pre- 
fixing 8  times  (8  +  1)  to  4  times  6,  7224. 

22.  62x68  26.  5.5x5.5  30.  46x44 

23.  74  X  76  27.  51  x  51  31.  3.7  x  3.3 

24.  83  X  87  28.  21  x  29  32.  95  X  95 

25.  55  X  55  29.  2.1  x  2.9  33.  71  X  71 


Division  273 

To  multiply  a  number  by  100,  1000,  etc.,  annex  to  the  number  as 
many  cipliers  as  there  are  in  the  multiplier,  or  remove  the  decimal 
point  a  corresponding  number  of  places  to  the  right. 

34.  486  X  1000  37.    48.6  x  1000  40.    100  x  2.432 

35.  243  X  2000  38.    .0486  X  100         41.    2000  x  2.43 

36.  4.86  X  1000         39.    1000  x  3.75         42.   300  x  .123 

DIVISION. 

585.  Division  is  the  process  of  obtaining  a  number  called 
the  quotient  by  performing,  on  a  number  called  the  dividend, 
the  operation  that  must  be  performed  upon  the  divisor  to 
produce  unity. 

586.  If  horses  are  worth  $135  each,  the  cost  of  27  horses 
is  found  by  multiplication  to  be  $3645  (S135  x  27).  Two 
different  problems  in  division  may  arise  from  this  example 
in  multiplication: 

1.  At  $135  each,  how  many  horses  can  be  bought  for 
$3645? 

2.  If  27  horses  cost  $  3645,  what  is  the  cost  of  one  horse  ? 

In  the  first  example  it  is  necessary  to  separate  $3645 
into  parts,  each  containing  $135  ;  in  the  second  example  it 
is  necessary  to  separate  $3645  into  27  parts. 

$3645  ^  $135  =  27,  the  number  of  horses. 
$3645  -V-  27  =  S135,  the  cost  of  one  horse. 

Although  different  names  are  given  to  these  different 
examples  in  division,  the  process  is  the  same. 

587.  When  the  dividend  and  the  divisor  are  like  concrete 
numbers,  the  quotient  is  abstract.  AYhen  the  dividend  is 
concrete  and  the  divisor  is  abstract,  the  quotient  is  concrete, 
having  the  same  unit  as  the  dividend. 


274  Arithmetic 

588.  Divide  365,841  by  482. 

.        TTo  3  Since  the  divisor  is  an  integer,  each  partial  quo- 

AllS^         i  •)>'-T^2-  .  .  p         ,  IP-  1  ■      1 

.  tient  IS  of  the  same  order  of  units  as  tlie  partial 

dividend  used  to  obtain  it.     The  first  partial  divi- 
3374 

dend  that  will  contain  the  divisor  is  3658  hundreds, 

2844 

which  contains  the  divisor  7  times.     Place  the  7  in 
941 0 
— - —  the  hundreds'  place  (just  above  the  8  of  the  partial 

dividend)  and  multiply  the  divisor  by  it,  obtaining 
-^-^  3374  hundreds.     Subtracting  this  product  from  the 

partial  dividend  gives  a  remainder  of  284  hundreds, 
or  2840  tens.  Bring  down  4  tens  from  the  dividend,  making  a  new 
partial  dividend  of  2844  tens.  Proceeding  as  before,  the  next  quotient 
figure  is  5  tens,  and  the  new  remainder  is  434  tens,  or  4340  units. 
Bring  down  1  unit  from  the  dividend,  making  a  new  partial  dividend 
of  4341  units.  The  next  quotient  figure  is  9  units,  and  the  remainder 
is  3  units.  The  remainder  is  written  as  the  numerator  of  a  fraction, 
having  the  divisor  482  as  the  denominator, 

589.  Proof.  The  correctness  of  the  result  may  be  tested 
by  adding  the  remainder  to  the  product  of  the  divisor  and 
the  quotient.  If  the  sum  is  equal  to  the  dividend,  the  work 
is  correct. 

.  590.  Division  of  Decimals. 

Divide  .872355  by  5.8157. 

Ans.    .15  Make  the  divisor  a  whole  number  by  moving  the 

58157)8723.55  decimal  point  four  places  to  the  right,  and  make  a 

58157  corresponding  change  in  the  dividend. 
290785  The  number  of  places  in  the  result  will  be  equal 

290785  to  the  number  of  places  in  the  dividend. 

Note.  In  the  foregoing  example  the  divisor  and  the  dividend  were 
multiplied  by  the  same  number,  10,000,  which  does  not  change  the 
value  of  the  quotient. 

591.  Division  of  Fractions. 

1.    Divide  117^  by  f. 

1)117^  Multiplying  the  divisor  and  the  dividend  by  the 

X  4    X  4  denominator  of  the  fraction  in  the  divisor  gives  a 

3)470  whole  number  for  the  divisor. 

1501  Ans. 


Division  275 

2.    Divide  llTi  by  3|. 

jg.^3-  Change  the  dividend  to  the  improper  fraction 

?      ^-  ^1^  and  the  divisor  to  V.     To  divide  a  number 

X  o   X  5 

by  -y-  the  divisor  is  multiplied  by  5,  givinn;  18  as 

the  new  divisor,  and  the  dividend  is  multiplied 

by   5.     The   quotient   is   5   times   the   dividend 

divided  by  18. 


18)^1^  X  5 
^P  X  5  -  18 

or,  2  35    X    jT"^. 


An  examination  of  the  foregoing  examples  will  show  that 
in  each  case  there  has  been  performed  on  the  dividend  the 
operation  that  mnst  be  performed  on  the  divisor  in  order  to 
produce  unity.  In  the  first,  the  dividend  has  been  multi- 
plied by  "I ;  in  the  second  by  y^g.  Each  of  these  multipliers 
is  the  reciprocal  of  the  corresponding  divisor ;  that  is,  it  is 
the  divisor  inverted. 

592.  To  divide  one  fraction  by  another  invert  the  divisor 
and  proceed  as  in  multiplication. 

Note.    Mixed  numbers  should  be  changed  to  improper  fractions. 

593.  Divide  1872f  by  11. 

11)1872|  When  the  divisor  is  an  integer  less  than  13,  the  result 

170^1        should  be  obtained  by  short  division.     Here  the  quotient 
is  170  with  a  remainder  of  2f ,  or  Jj^  one  eleventh  of  which  is  ^|. 

594.  Oompound  Division. 

1.  Divide  18  gal.  1  pt.  by  5. 

Insert  0  qt.    18  gal.  -^  5  =  3  gal. ,  remainder 
5)18  gal.   0  qt.  1  pt.         3  gal.,  or  12  qt.     12  qt.  -^  5  =  2  qt.,  remain- 
8  gal.  2  qt.  1  pt.         der  2  qt.,  or  4  pt.       4  pt.  +  1  pt.  =5  pt. 
5  pt.  H-  5  =  1  pt. 

2.  Divide  18  gal.  1  pt.  by  3  gal.  2  qt.  1  pt. 

3  gal.  2  qt.  1  pt.  =  29  pt.      18  gal.  1  pt.  =  145  pt. 

on    ^  M  4 '    ^  /c  The  dividend  and  the  divisor  must  be  of 

29  pt.)14Dpt.  (5  .       ^    ,        ,     , 

,,_    ^  the  same  denomniate  unit.     Reduce  both  to 

14o  pt. 

pints.     145  pt.  -=-  29  pt.  =  5. 


276  Arithmetic 

595.  Ratio, 

Ratio  is  the  relative  value  that  exists  between  any  two 
unequal  like  numbers.     It  is  determined  by  division. 

The  ratio  of  15  to  45  is  1  ;  that  is,  15  -=-  45,  or  i|  reduced  to  lowest 
terms.  The  ratio  of  3  gal.  2  qt.  1  pt.  to  18  gal.  1  pt.  is  the  ratio  of 
29  pt.  to  145  pt.,  or  j\%,  or  ^. 

596.  Only  numbers  of  the  same  unit  value  can  be 
compared. 

597.  Oancellation. 

Operations  involving  only  multiplication  and  division  can 
be  shortened  by  canceling  factors  common  to  the  divisor 
and  to  the  dividend. 

598.  Divide  40  x  3  x  1.89  x  12  by  6.3  x  4  x  72.      . 

^  Write   the    factors    composing 

X^  .3  the  dividend  above  the  line  and 

^0  X  3  X  l^.^  X  12  _  r       q  _  1  K       those  composing  tlie  divisor  below, 
63  X  ^  X  J^  *      .     '  '      first  changing  6.3  to  a  whole  num- 

^  ber  and  making  a  corresponding 

^  change  in  1.89,  the  latter  becom- 

ing 18.9.  Cancel  63  and  18.9,  writing  .3  over  the  latter.  Cancel  12 
and  72,  writing  6  under  the  latter.  Cancel  3  and  6,  writing  2  under 
the  latter.  Cancel  4  and  40,  writing  10  over  the  latter.  Cancel  2  and 
10,  writing  5  over  the  latter.  The  remaining  factors  are  5  and  .3  in 
the  dividend.     Their  product,  1.5,  is  the  result. 

599.  Multiplying  or  dividing  the  divisor  and  the  divi- 
dend by  the  same  number  does  not  change  the  value  of 
the  quotient. 

MISCELLANEOUS   PROBLEMS. 

600.  Oral  Problems. 

1.    What  is  the  ratio  of  |  to  f  in  whole  numbers  ? 
'  2.    What  part  of  a  cubic  foot  is  a  board  foot  ? 
3.    Into  how  many  cubes  J  ft.  long  can  a  cubic  foot  be  cut  ? 


Miscellaneous   Problems  277 

4.  One  half  of  A's  money  equals  one  third  of  B's.  B  has 
$  1000  more  than  A.     How  much  has  each  ? 

5.  What  is  the  date  of  maturity  of  a  90-day  note  drawn 
Sept.  7  ? 

6.  How  many  gallons  of  water  must  be  mixed  with  76 
gallons  of  vinegar  so  that  the  mixture  will  contain  5%  of 
water  ? 

7.  Find  the  cost  at  1^  per  sq.  ft.  of  painting  a  church 
spire  whose  base  is  a  hexagon,  each  side  measuring  10  feet, 
and  the  slant  height  being  60  feet. 

8.  A  dealer  bought  200  barrels  of  flour  at  $5  per  bbl. 
He  sells  50  barrels  that  were  damaged  for  f  4  per  bbl.,  and 
the  remaining  150  bbl.  at  $6  per  bbl.  What  per  cent  does 
he  gain  on  the  entire  transaction? 

9.  A  boy  buys  apples  at  5  for  3/  and  sells  them  at  4  for 
3)^.    How  many  has  he  sold  if  his  profit  is  $  3  ? 

10.  Three  fourths  of  the  value  of  A's  house  is  equal  to 
two  thirds  of  the  value  of  B's.  What  is  the  ratio  of  the 
value  of  A's  house  to  that  of  B's  ? 

11.  What  is  the  interest  on  f  600  at  4%  for  270  days? 

12.  If  the  side  of  one  square  lot  is  5  rods,  and  that  of 
another  is  10  rods,  how  do  the  lots  compare  in  area  ? 

13.  Three  fourths  of  M's  farm  is  equal  in  value  to  two 
thirds  of  N's.  Together  the  farms  are  worth  f  17,000. 
What  is  the  value  of  each  ? 

14.  A  rectangular  lot  measuring  6  rods  on  one  side  has  a 
diagonal  of  10  rods.     What  is  the  area  ? 

15.  How  many  square  rods  in  a  field  in  the  form  of  a 
trapezoid  having  parallel  sides  of  50  and  40  rods  respectively 
and  an  altitude  of  45  rods  ? 

16.  Three  fourths  per  cent  of  A's  money  is  $  60.  How 
much  money  has  he  ? 


278  Arithmetic 

17.  A  man  is  engaged  for  a  year  for  f  280  and  a  suit  of 
clothes.  At  the  end  of  6  months  he  receives  $  130  and  the 
clothes.     What  is  the  value  of  the  suit  ? 

18.  Find  the  cost  of  88  lbs.  of  coffee  at  24|  ^  per  lb. 

19.  If  goods  cost  $  240  and  the  j^rofit  on  them  is  125%, 
what  is  the  selling  price  ? 

20.  Find  the  value  of  a  pile  of  wood  8  ft,  by  4  ft.  by  4 
ft.,  at  $  4  per  cord. 

21.  A  merchant  buys  goods  at  $  2  per  yard ;  he  sells  them 
10%  below  the  marked  price  and  still  makes  a  profit  of 
12i%.     Find  the  selling  price.     The  marked  price. 

22.  In  what  time  at  6%  will  f  100  amount  to  $  106.60  ? 

23.  A  person  bought  stock  at  20%  above  its  face  value 
and  sold  it  at  10%  below  its  face  value.  What  per  cent  of 
the  sum  invested  did  he  lose  ? 

24.  What  single  discount  is  equal  to  successive  discounts 
of  40  and  10%  ? 

25.  After  spending  |-  of  his  money  and  Jq-  of  the  remain- 
der a  boy  has  54  cents.     How  much  had  he  at  lirst  ? 

26.  A  man  pays  60  cents  for  an  article  at  a  deduction  of 
40  and  10%  from  the  list  price.     What  was  the  list  price? 

.  27.  How  many  square  yards  of  fencing  6  feet  high  will 
be  required  to  build  a  tight  board  fence  about  a  rectangular 
plot  150  feet  long  by  120  feet  wide  ? 

28.  In  what  time  will  a  sum  of  money  double  itself  at 
41%  simple  interest? 

29.  Divide  If  by  f. 

30.  How  many  acres  are  there  in  a  field  20  rd.  x  32  rd.  ? 

31.  A  house  is  sold  for  $4000,25%  of  which  is  profit. 
What  per  cent  would  have  been  gained  had  the  house  been 
sold  for  $  3360  ? 


Miscellaneous   Problems  279 

32.  What  is  the  loss  on  100  shares  of  stock  bought  at  90 
and  sold  at  88,  brokerage  i%  for  buying  and  i%  for  selling  ? 

33.  What  is  the  profit  on  100  shares  of  stock  bought  at 
88  and  sokl  at  90,  brokerage  -i-%  for  buying  and  -J-%  for 
selling  ? 

34.  How  many  gallons  of  ice  cream  will  be  needed  to 
supply  128  people  with  ^  pint  each  ? 

35.  A  rectangular  field  is  99  yards  long  and  87  yards 
wide.     How  many  square  yards  does  it  contain  ? 

36.  What  principal  will  produce  $  150  interest  in  2  yr. 
6  mo.  at  6%  ? 

37.  How  many  rectangular  blocks  8  in.  x  4  in.  x  2  in. 
will  be  required  to  build  a  block  8  ft.  x  4  ft.  x  2  f t.  ? 

38.  Find  the  quotient  of  302y\  divided  by  10. 

39.  Multiply  20^  by  6|-. 

40.  How  many  hours  in  April  ?  In  March  ?  In  Feb- 
ruary, 1908  ?     In  February,  1909  ? 

41.  A  house  costing  $  4000  rents  for  $  30  per  month.  If 
the  expenses  for  repairs,  taxes,  etc.,  are  $  GO  per  year,  what 
per  cent  does  the  owner  make  ? 

42.  The  ratios  at  which  A,  B,  and  C  work  are  to  each 
other  as  2,  3,  and  4.  What  whole  numbers  will  represent, 
respectively,  the  time  taken  by  each  to  do  a  piece  of  work  ? 

43.  Find  the  area  of  a  plot  of  ground  9|  yards  long  by  9^ 
yards  wide. 

44.  What  sum  invested  in  6%  bonds  at  150  will  yield 
$  COO  per  year  ? 

45.  Three  fourths  is  what  per  cent  of  |  ? 

46.  By  what  per  cent  is  the  value  of  the  fraction  f 
increased  when  G  is  added  to  the  numerator  and  to  the 
denominator  ? 


2  8o  Arithmetic 

47.  At  what  rate  of  interest  will  $100  in  3  yr.  4  mo. 
amount  to  $  115  ? 

48.  A  woman  spends  ^  of  her  money  for  a  hat,  |  of  it  for 
a  dress,  and  has  $  12  remaining.     What  had  she  at  first  ? 

49.  At  the  rate  of  371  cents  a  pint,  how  many  gallons 
of  olive  oil  can  be  purchased  for  $15? 

50.  The  volume  of  a  cube  is  1728  cubic  inches ;  how  many 
square  inches  in  its  entire  surface? 

51.  Find  the  proceeds  of  a  60-day  note  for  $250,  dis- 
counted the  day  it  is  drawn  at  6 %. 

52.  What  number  divided  by  -f-  gives  a  quotient  of  -|  ? 

,  53.    How  many  pieces  of  ribbon  each   containing   -J  yd. 
can  be  cut  from  a  piece  containing  56  yd.  ? 

54.  Find  the  volume  of  a  square  pyramid  12  inches 
high,  each  side  of  the  base  measuring  5^  inches. 

55.  I  sold  a  carriage  for  $  240,  on  which  I  lost  20  %  ; 
what  did  I  lose  ?     What  fraction  of  $  240  did  I  lose  ? 

56.  What  did  I  gain  on  a  carriage  sold  for  $  240,  on 
which  my  profit  was  20  %  ?  What  fraction  of  $  240  did  I 
gain  ? 

57.  Tea  costing  20  cents  a  pound  is  mixed  with  an  equal 
quantity  of  tea  costing  30  cents  a  pound.  What  per  cent 
is  gained  by  selling  the  mixture  at  35  cents  per  pound  ? 

58.  A  can  do  a  piece  of  work  in  3  days  and  B  can  do  it  in 
6  days.  They  work  together  and  get  $  12  for  the  work. 
How  much  should  each  receive?  How  many  days  do  they 
take  ? 

59.  What  will  it  cost  to  carpet  a  room  18  ft.  long,  15  ft. 
wide  at  f  .75  per  yard,  the  carpet  being  J  yd.  wide  ? 

60.  A  register-24  inches  by  12  inches  admits  hot  air  to  a 
room.  Give  the  dimensions  of  a  register  that  is  double  the 
size. 


Miscellaneous   Problems  281 

61.  Give  the  inside  dimensions  of  a  box  tliat  will  exsLCtly 
contain  1  gallon  (231  cubic  inches). 

62.  Divide  8.4  by  .4. 

63.  A  quantity  of  sirup  that  has  lost  20%  by  leakage 
is  sold  by  the  gallon  at  40  %  above  cost.  What  is  the  gain 
per  cent  ? 

64.  AVhat  is  the  cube  root  of  729  ? 

65.  Reduce  9J  to  an  improper  fraction,  giving  the  reason 
for  each  successive  step.     . 

66.  Multiply  66  by  64. 

67.  In  how  many  years  and  months  will  any  sum  double 
itself  at  6  %  simple  interest  ? 

68.  Divide  f  2100  among  three  persons  so  that  the  sec- 
ond may  receive  one  half  as  much  as  the  first,  and  the 
third  one  half  as  much  as  the  second. 

69.  In  the  number  72,930,  the  local  value  expressed  by 
the  first  two  digits  is  how  many  times  the  local  value 
expressed  by  the  second  digit  ?  By  the  third  digit  ?  By 
the  fourth  digit  ? 

70.  Find  the  cost  of  a  two-inch  plank  18  ft.  long,  8  in. 
wide,  at  $  40  per  M. 

71.  The  capacity  of  a  bin  is  40  bu.  of  1^  cu.  ft.  What 
is  the  depth  of  the  bin,  if  the  length  is  5  ft.,  and  the  width 
4ft.? 

72.  How  much  coal  at  $6  per  ton  is  equal  in  value  to 
1800  lb.  hay  at  f  12  per  ton  ? 

73.  A  dealer  paid  $950  for  200  barrels  of  flour.  In 
addition  he  pays  20  ^  per  barrel  for  freight,  and  5  ^  per 
barrel  for  cartage.  What  must  be  the  selling  price  of  the 
flour  per  barrel,  to  produce  a  profit  of  10%  on  the  total 
cost? 


282  Arithmetic 

601.    Written  Problems. 

1.  If  a  six-inch  cube  of  stone  weighs  21  pounds,  what 
will  be  the  weight  of  a  cubic  foot  ?  How  many  times  as 
heavy  as  water  is  the  stone,  a  cubic  foot  of  water  weighing 
1000  ounces  ? 

2.  Express  in  whole  numbers  the  ratio  of  -||-  to  -|-^. 

3.  What  rate  of  interest  do  I  realize  on  stock  paying 
6%  dividends,  if  it  cost  me  108,  including  brokerage  ? 

4.  Find  the  volume  of  a  cube,  whose  entire  surface  con- 
tains 100.86  square  inches. 

5.  I  owe  a  bill  amounting  to  $311.85,  for  the  payment 
of  which  I  draw  a  60-day  note.  What  must  be  the  face  of 
a  note  which  will  yield  the  amount  due,  if  discounted  ? 

6.  What  is  the  diameter  of  a  circular  piece  of  ground 
that  will  contain  8  acres  106  square  rods  ? 

7.  Add  :  123,456 :  78,695  ;  57,989 ;  6078  ;  85,769  ;  8888 ; 
67,756  ;  200,009  ;  77,777  ;  85,685 ;  9999  ;  8,476,780  ;  57,869  ; 
308,705;  96,878;  9586;  888;  57;  9. 

8.  What  principal  will  produce  $  1339.31  interest  in 
2yr.  7  mo.  24  da.  at  7%  ? 

9.  Sold  I  of  an  article  for  what  f  of  it  cost.  What  was 
the  gain  per  cent  ? 

10.  What  was  the  gain  per  cent  in  the  sale  of  a  horse 
when  I  of  the  gain  equaled  |  of  the  cost  ? 

11.  How  many  cubic  feet  of  water  will  be  discharged 
in  5  hours  from  a  pipe  2  feet  in  diameter,  if  the  water  flows 
at  the  rate  of  10  miles  per  hour  ? 

Volume  of  cylinder  2  feet  in  diameter,  50  miles  high. 

12.  How  many  square  feet  in  a  plow  furrow  1  mile  long, 
f  foot  wide  ?  How  many  such  furrows  will  contain  an 
acre  ? 


Miscellaneous    Problems  283 

13.  By  selling  goods  at  46^  per  yard  a  loss  of  8%  :s 
incurred.  How  much  must  this  selling  price  be  increased 
in  order  that  a  profit  of  15%  may  be  realized? 

14.  By  what  per  cent  is  the  value  of  the  fraction  i|  in- 
creased, when  6  is  added  to  the  numerator  and  to  the 
denominator  ? 

15.  If  1800  sq.  ft.  of  boards  are  needed  to  fence  a  lot 
30  ft.  by  120  ft.,  how  many  sq.  ft.  will  be  needed  to  fence 
a  lot  25  ft.  by  100  ft.? 

16.  What  is  the  diagonal  of  a  square  field  containing 
20  acres? 

17.  It  costs  8.25  francs  to  ship  3000  kilos  10  kilometers. 
What  will  it  cost  to  ship  2400  kilos  62.5  kilometers? 

18.  A  pile  of  4-foot  wood  36  ft.  long  is  12  ft.  high  at  one 
end,  and  slopes  regularly  to  a  height  of  8  ft.  at  the  other ; 
how  many  cords  does  it  contain? 

19.  Divide  $150  between  A  and  B  so  that  A's  share  may 
be  to  B's  share  as  ^  to  ^. 

20.  If  3.5  yd.  of  cloth,  1.75  yd.  wide,  will  make  a  suit 
of  clothes,  how  much  would  be  required  if  the  cloth  were 
only  f  yd.  wide  ? 

21.  The  surface  of  a  cube  is  486  sq.  in. ;  what  is  the 
surface  of  a  cube  one  third  the  length  of  the  first  ? 

22.  A  cubical  block  of  marble  contains  27  cubic  feet; 
how  many  cubic  feet  are  there  in  a  second  cubical  block 
having  three  times  the  dimensions  of  the  first? 

23.  If  2  yd.  1  ft.  2  in.  of  ribbon  cost  43  cents,  what  would 
have  to  be  paid  for  9  yd.  2  ft.  10  in.  ? 

24.  I  gain  12%  by  selling  silk  for  $  1.68  per  yard.  What 
should  be  my  selling  price  to  gain  24%  ? 

25.  A  horse  is  fastened  to  a  stake  in  the  center  of  a  field 
by  a  rope  75  feet  long.  If  he  can  reach  2  feet  beyond  the 
rope,  what  fraction  of  an  acre  can  he  graze  oyer  ? 


284  Arithmetic 

26.  A  man  is  engaged  for  a  year  for  $  300  and  a  suit  of 
clothes.  He  receives  the  latter  at  once,  and  at  the  end  of 
7  months  he  leaves,  receiving  $165  in  cash.  What  is  the 
value  of  the  clothes  ? 

27.  What  must  be  the  face  of  a  note  that  will  yield  pro- 
ceeds of  $240,  when  discounted  at  a  bank  for  69  days  at 
6%  ? 

28.  A  stick  of  timber  is  16  inches  broad  and  10  inches 
thick;  what  length  of  it  will  make  10  cubic  feet? 

29.  I  invest  $19,950  in  4  per  cents  at  95,  including 
brokerage.     What  annual  income  do  they  yield? 

30.  A  floor  24  ft.  2  in.  long  and  18  ft.  11  in.  wide  is 
covered  with  carpet  f  yd.  wide,  costing  $  1  per  yard.  How 
much  will  be  saved  by  running  the  strips  the  more  eco- 
nomical way? 

31.  A  merchant  sold  20  hhds.  of  olive  oil,  each  contain- 
ing 63  gallons,  at  $1.75  per  gallon,  and  invested  the  pro- 
ceeds in  table  sauce  in  cases  of  12  bottles  each,  worth  $0.25 
a  bottle.  How  many  cases  did  he  buy?  (Solve  by  cancella- 
tion.) 

32.  I  bought  a  house  and  paid  \  of  the  cost  down,  and  -| 
of  the  remainder  at  the  end  of  the  year.  The  two  payments 
amounted  to  $  9600.     What  was  the  cost  of  the  house  ? 

33.  Multiply  2.4698  by  1000,  and  divide  64.2  by  10,000, 
by  removing  the  decimal  point  in  each  case.  Explain  each 
operation. 

34.  If  7  bu.  2  pk.  7  qt.  of  chestnuts  cost  $14.85,  what 
are  the  chestnuts  worth  per  bushel  ? 

35.  Which  is  the  better  discount  on  a  purchase  of  dry 
goods  amounting  to  $  1875.50,  40  and  10%,  or  30  and  20%  ? 
What  is  the  difference? 

36.  A  gentleman  wishes  to  endow  a  professorship  in  a 
college  with  an  annual  income  of  $6000.     What  amount 


Miscellaneous   Problems  285 

must   he   invest   for   that  purpose  in  4%   stocks  at  122 J, 
brokerage  |^%  ? 

37.  A  debt  which  I  incurred  5  yr.  3  mo.  15  da.  ago, 
amounts,  with  interest  at  7%,  to  $2548.975.  What  was  the 
original  debt? 

38.  I  own  65%  of  a  mill  and  sell  40%  of  my  share  for 
$5538.     At  that  rate  what  is  the  mill  worth? 

39.  A  square  prism  28  inches  high  contains  112  cu.  in. 
How  many  inches  square  is  it  at  either  end  ? 

40.  A  certain  field  is  in  the  form  of  a  trapezoid.  Its 
parallel  sides  measure  respectively  22.4  rods  and  35.2  rods, 
and  the  perpendicular  distance  between  them  is  38.4  rods. 
What  is  the  value  of  the  field  at  $  160  an  acre? 

41.  Show  why  removing  a  decimal  point  three  places  to 
the  left  divides  by  1000. 

42.  Bought  3  hhd.  15  gal.  3  qt.  catsup  at  the  rate  of  $42 
per  hogshead  of  63  gal.  If  I  bottle  it  in  three-pint  bottles 
costing  5  cents  each,  at  how  much  a  bottle  must  I  sell  it  so 
that  I  may  neither  gain  nor  lose? 

43.  How  many  yards  of  paper  30  inches  wide  will  be  re- 
quired to  cover  the  Avails  of  a  room  15  ft.  long,  11  ft.  wide, 
and  7  ft.  high,  making  no  allowance  for  openings  ? 

44.  What  part  of  -^  of  an  acre  is  j%  of  a  sq.  rd.  ? 

45.  Find  the  sum  of  the  following  distances :  16  mi.  298 
rd.  14  ft.,  19  mi.  53  rd.  16  ft,  97  mi.  147  rd.  13  ft.,  and 
47  mi.  237  rd.  13  ft. 

46.  A  merchant  sold  a  piece  of  cloth  for  $33  and  lost  8|- 
per  cent.  What  per  cent  would  he  have  gained,  had  he  sold 
it  for  $43? 

47.  («)  At  5  per  cent,  find  the  present  worth  and  the 
true  discount  of  $126.06,  due  in  8  mo.  12  da.  (b)  Re- 
garding the  foregoing  problem  as  a  problem  in  interest,  tell 


286  Arithmetic 

which  of  the  terms  used  in  interest  are  given,  and  which  are 
required. 

48.  At  what  price  must  I  buy  stocks  yielding  an  annual 
dividend  of  41  per  cent,  that  my  investment  may  pay  me 
3  per  cent  interest  ? 

49.  If  the  services  of  eight  men  8  hours  a  day  for  six 
days  are  worth  $  132,  what  should  be  paid  for  the  services 
of  six  men  10  hours  a  day  for  eight  days? 

60.  A  grocer  mixed  8  lb.  coffee  worth  25  cents  a  pound 
with  12  lb.  worth  30  cents  a  pound.  At  what  price  should 
he  sell  the  mixture  so  that  he  may  neither  gain  nor  lose? 

51.  The  large  wheels  of  an  engine  are  16  feet  6  inches  in 
circumference.  How  many  revolutions  will  one  of  them 
make  in  traveling  5  miles  ? 

52.  A  boy  laid  away  15  cents  a  day,  Sundays  included. 
What  did  his  savings  amount  to  from  June  17,  1907,  to 
May  12,  1908  ? 

53.  From  a  piece  of  land  containing  75  A.,  I  have  sold 
ten  lots  each  30  rd.  square.  What  is  the  remaining  land 
worth  at  $  25.60  an  acre  ? 

54.  A  gentleman  left  f  of  his  estate  to  his  wife,  ^  of  the 
remainder  to  his  son,  and  |  of  what  then  remained  to  his 
daughter,  who  received  $  750.  What  was  the  value  of  the 
whole  estate? 

55.  What  number  must  be  multiplied  by  .085  that  the 
product  may  be  1450.1  ?  What  number  must  be  divided  by 
.032  that  the  quotient  may  be  212.6  and  the  remainder  .0008  ? 

56.  A  note  for  ^325  was  dated  June  5,  1908.  The  rate 
of  interest  being  5J  per  cent,  and  no  payment  having  been 
made  on  the  note,  how  much  was  due  at  settlement,  July 
5,  1909  ? 

57.  Bought  87  sliares  in  a  mining  company  at  12  per  cent 
below  par,  and  sold  the  same  at  19|^  per  cent  above  par. 


Miscellaneous   Problems  287 

What  sum  did  I  gain,  the  par  value  of  a  share  being  $  75  ? 
No  brokerage. 

58.  A  bankrupt's  assets  are  ^45,000,  and  his  liabilities 
are  $67,500.  How  much  can  he  pay  on  the  dollar;  and 
how  much  should  A,  to  whom  he  owes  $12,675,  receive? 

59.  Bought  a  hogshead  of  vinegar  for  $21.  Having  lost 
lOJ^  gallons  by  leakage,  at  what  price  per  gallon  must  I  sell 
the  remainder  to  gain  25  per  cent  on  the  whole?  (1  hogs- 
head =  63  gallons.) 

60.  It  is  proposed  to  exchange  a  tank  9  ft.  square  and  li- 
ft, deep  for  another  of  equal  capacity,  but  cubical  in  form. 
What  will  be  the  dimensions  of  the  latter  in  feet  and  inches? 

61.  Find  the  value  in  pounds,  shillings,  and  pence,  of 
$247.59,  the  pound  sterling  being  $4.8665. 

62.  New  York  is  74°  3'  west  of  London.  When  it  is  4.30 
A.M.,  solar  time,  at  New  York,  what  is  the  time  at  London? 

63.  A  pound  avoirdupois  contains  7000  troy  grains.  At 
80^  per  ounce  troy,  what  is  the  value  of  a  silver  pitcher 
weighing  1  lb.  8  oz.  avoirdupois  ? 

64.  A  man  bought  43.75  yards  of  carpet  for  $58|^;  he 
sold  f  of  it,  gaining  $  .16|  on  each  yard  sold.  How  much 
did  he  receive? 

65.  A  merchant,  places  a  bill  of  $840  in  the  hands  of  a 
collector,  who  succeeds  in  obtaining  75  per  cent  of  it,  and 
charges  5  per  cent  commission.  How  much  does  the  mer- 
chant receive? 

66.  It  is  80  rods  between  the  diagonally  opposite  corners 
of  a  square  field ;  how  many  acres  in  the  field  ? 

67.  A  grocer  mixes  150  pounds  of  black  tea  with  80 
pounds  of  green  tea,  and  gains  38  per  cent  by  selling  the 
mixed  tea  for  30  cents  per  pound.  If  the  black  tea  costs  20 
cents  a  pound,  what  is  the  cost  per  pound  of  the  green  tea  ? 


288  Arithmetic 

68.  How  much  profit  is  made  on  3600  meters  of  silk 
bought  at  $1.60  per  meter  and  sold  at  $1.75  per  yard? 
(A  meter  is  39.37  inches.) 

69.  A  commission  merchant  sells  for  a  farmer  45  tubs  of 
butter  containing  40  pounds  each,  for  18f  cents  per  pound, 
charging  him  3  per  cent  commission.  With  the  proceeds  he 
buys  groceries,  charging  3  per  cent  for  buying.  AVhat  is  his 
commission  on  the  sale,  his  commission  on  the  purchase,  and 
the  cost  of  the  groceries  ? 

70.  In  building  a  concrete  wall,  2  parts  of  lime,  1  part  of 
cement,  and  6  parts  of  broken  stone  are  used.  How  many 
cubic  feet  of  each  are  required  in  a  wall  36'  x  9'  x  1^'? 

71.  Solve  the  following  question  by  cancellation.  How 
many  rolls  of  merino,  each  containing  75  yd.,  worth  $  .45  a 
yd.,  will  it  take  to  pay  for  180  yd.  of  alpaca  at  $.30  a  yd.  ? 

72.  Paid  $487|-  for  a  lot  of  apples,  at  the  rate  of  $.60  for 
I  of  a  barrel.     How  many  barrels  did  I  buy? 

73.  The  difference  in  solar  time  between  two  places,  A 
and  B,  is  5  hr.  5  min.  20  sec,  A  having  the  earlier  time.  If 
B  is  18°  west  longitude,  what  is  the  longitude  of  ^? 

74.  Sold  certain  goods  for  $128.05,  gaining  $9.85.  What 
was  the  %  of  gain? 

75.  Find  the  face  of  a  sight  draft  on  St.  Louis  that  may 
be  bought  for  $350,  exchange  being  at  $1  premium  per 
$  1000. 

76.  I  have  $5000  in  savings  banks,  drawing  interest  at 
4%.  If  I  invest  it  in  6%  stocks  at  125  including  brokerage, 
how  much  more  shall  I  receive  annually  than  if  I  had  kept 
it  in  the  bank? 

77.  When  may  $600  due  in  4  mo.,  $800  due  in  5  mo.,  and 
$  1000  due  in  10  mo.,  be  paid  at  one  time  without  loss  to 
either  debtor  or  creditor? 


Miscellaneous  Problems  289 

78.  A  cylindrical  cistern  contains,  when  full,  3080  cu.  ft. 
of  water.     Its  depth  is  20  ft.     What  is  its  diameter  ? 

79.  The  entire  surface  of  a  certain  cube  is  1014  sq.  in. 
What  is  its  volume? 

80.  Bought  140  yards  of  cloth  at  $1.65  per  yard.  Sold 
I  of  it  at  $1.95  per  yard.  At  what  price  per  yard  must  the 
remainder  be  sold  to  gain  10  per  cent  on  the  whole  ? 

81.  The  product  of  three  factors  is  14.  One  of  the 
factors  is  4;  the  remaining  two  are  equal.  What  are  the 
equal  factors? 

82.  What  per  cent  do  I  gain  or  lose  by  deducting  20  per 
cent  from  the  price  of  goods  marked  20  per  cent  above  cost? 

83.  In  what  time  will  $1395  amount  to  $1431.27  at  4 
per  cent? 

84.  Express  in  pounds  and  fraction  of  a  pound  the  weight 
of  a  pint  of  water.  (A  gallon  contains  231  cubic  inches.  A 
cubic  foot  of  water  weighs  1000  ounces.) 

85.  One  side  of  a  rectangular  field  containing  21|-  acres, 
measures  180  rods.  What  is  the  length  of  the  diagonal  of 
the  field? 

86.  The  population  of  a  certain  city  was,  in  1900, 14,500 ; 
in  1910,  124,250.     What  is  the  percentage  of  increase? 

87.  A  man  bequeathed  one  third  of  his  property  to  his 
wife,  one  fourth  of  the  remainder  to  each  of  three  daughters, 
the  rest  to  his  son.  The  difference  between  his  wife's  share 
and  that  of  his  son  was  $6750.75.  How  much  did  each 
receive  ? 

88.  Which  is  the  greater  discount,  40  and  20  per  cent  or 
50  and  10  per  cent  ?     Why  ? 

89.  A  bar  of  iron  is  6  feet  long,  15  inches  wide,  and  10 
inches  thick.  How  much  is  its  length  increased,  when  the 
width  is  reduced  by  rolling  to  5  inches  and  the  thickness  to 
4  inches,  the  bulk  remaining  the  same  ? 


290  Arithmetic 

90.  A  garden  25  ft.  by  120  ft.  is  to  be  surrounded  by  a 
brick  wall  5  ft.  high  and  12  inches  thick.  How  many  bricks 
will  be  required  if  22  bricks  with  the  mortar  will  lay  one 
cubic  foot  ? 

Outside  measurement  of  garden  and  wall,  27'  x  122'  ;  inside  meas- 
urement, 25'  X  120'.     Difference  =  area  of  base  of  wall. 


91-    (2f^gi)+l(6A-2|)x|j=? 


92.  Write  an  analysis  of  the  work  required  to  solve  the 
question :  An  officer  in  pursuit  of  a  thief  runs  8  yards  to 
the  thief's  5,  but  the  latter  has  60  yards  start.  How  far 
does  the  officer  run  to  overtake  the  thief  ? 

93.  Divide  twenty-five  ten-thousandths  by  125,  multiply 
the  quotient  by  five  tenths,  and  divide  1000  by  the 
product. 

94.  If  12  men  can  saw  45  cords  of  wood  in  3  days,  work- 
ing 9  hours  a  day,  how  much  can  4  men  saw  in  18  days, 
working  8  hours  a  day  ? 

95.  What  per  cent  is  gained  or  lost  by  selling  f  of  an 
article  for  the  cost  of  4  of  it  ? 

96.  Erastus  Brooks  owes  me  $4489.32.  He  gives  mo 
his  note,  non-interest-bearing,  at  90  days.  The  note  dis- 
counted at  once,  at  a  bank,  at  6  per  cent,  yields  the  exact 
amount  of  the  debt.     Eequired  its  face. 

97.  I  once  lent  a  friend  $875  for  1  year  4  months. 
He    now^     proposes     to     lend     me 

$350  long  enough  to   balance  the  ^^^"" 

obligation.      Wliat    will    the    time 

be? 

98.  Find  the  area  in  square  feet 
of  the  field  represented  by  the 
following  diagram: 


Miscellaneous   Problems 


291 


99.    A  rectangular  field  whose  breadth  is  y\  of  its  length 
contains  3  acres.     What  is  the  length  of  the  diagonal? 

100.  At  75  cents  per  yard,  how  much  will  it  cost  to 
carpet  a  room  19  ft.  x  15  ft.  with  carpeting  |  yd.  wide,  the 
same  figure  recurring  at  intervals  of  8  inches  and  the  strips 
running  lengthwise  ? 

The  length  of  a  strip,  being  19  ft.,  contains  the  pattern  28  times  and 
4  inches  of  the  next  repetition.  To  match  at  the  top,  the  remaining 
4  inches  of  this  repetition  must  be  cut  off.  The  same  waste  occurs  in 
each  strip  except  the  first. 

101.  rind  the  least  common  multiple  of  35,  45,  63,  70. 

102.  Reduce  3.576  miles  to  its  equivalent  in  miles,  rods, 
and  yards. 

■  103.  If  coal  is  bought  at  $  6  per  ton  and  sold  at  the  rate 
of  30  cents  per  basket  of  80  pounds,  what  is  the  gain  per 
cent? 

104.  What  single  rate  of  discount  is  equal  to  successive 
discounts  of  25,  20,  and  5%? 

105.  Divide  64,564,000  by  798. 

106.  Reduce  to  its  simplest  form 

A  of  J5_  4-  i  of  -8- 
2.  of  -9 5.  of  -^ 

3   "^    1  4  6    '-'^    1  5 

107.  The  product  of  four  factors  is  432.  Two  of  the 
factors  are  3  and  4.  The  other  two  factors  are  equal. 
What  are  the  equal  factors  ? 

108.  What  per  cent  is  gained  by  selling  at  $  120  a  thou- 
sand articles  that  cost  $  9.50  a  hundred  ? 

109.  A  rectangular  room  measures  24  ft.  by  18  ft.  What 
will  it  cost  to  carpet  it  with  material,  three  quarters  of  a 
yard  wide,  at  $  1.12J  a  yard,  if  four  yards  are  wasted  in 
the  matching  ? 


292  Arithmetic 

110.  How  often  may  the  quotient  of  25  ten-thousandths 
-r- 125  be  subtracted  from  the  quotient  of  125  -7-  25  ten- 
thousandths  ? 

111.  A  commission  merchant  in  o^ew  Orleans  whose 
charge  for  buying  is  2\  per  cent,  retained  $  48  as  his  com- 
mission out  of  the  money  sent  him  for  a  purchase  of  sugar. 
What  was  the  amount  sent? 

112.  A  man  standing  150  feet  from  the  foot  of  a  tree 
50  feet  high,  shot  a  bird  hovering  over  the  top;  the  man 
was  170  feet  from  the  bird.  How  far  was  the  bird  from 
the  top  of  the  tree  ? 

113.  The  exact  interest  (365  days  to  the  year)  of  a  cer- 
tain sum  of  money  at  4  per  cent,  from  June  18  to  August 
30  (exact  number  of  days),  was  $40.  What  was  the  sum 
at  interest  ? 

114.  Trees  are  set  in  an  orchard  at  the  intersections  of 
lines  drawn  2  rods  apart.  If  the  outside  lines  are  each  1 
rod  from  the  boundaries,  how  many  trees  will  be  required 
in  a  3-acre  plot  24  rods  long? 

115.  Find  the  exact  interest  on  $475  from  July  16  to 
December  9  at  5%. 

116.  How  many  rods  of  fence  will  be  needed  to  inclose  a 
rectangular  40-acre  field,  one  side  of  which  measures  f  mile? 

117.  What  is  the  value,  at  $4  per  cord,  of  a  pile  of  wood 
16  feet  long,  4  feet  wide,  and  9|^  feet  high? 

118.  After  paying  an  agent  5%  of  the  sum  collected,  a 
man  has  $1436.40.  What  was  the  sum  collected  by  the 
agent  ? 

119.  A  certain  reservoir  will  hold  528,000  cubic  feet  of 
water.  What  must  be  the  size  of  a  square  pipe  which  will 
fill  the  reservoir  in  5  hours,  if  the  water  runs  at  the  rate  of 
10  miles  per  hour? 


Miscellaneous   Problems  293 

The  volume  of  water  to  flow  hourly  will  be  i  of  528,000  cu.  ft. 
This  may  be  considered  as  the  volume  of  a  square  prism  5  miles  high 
(26,400  ft.),  each  side  of  the  base  measuring  x  feet. 

120.  A  wagon  loaded  with,  hay  weighs  3175  lb.  The 
wagon  weighs  715  lb.  What  is  the  value  of  the  hay  at 
$15.75  per  ton? 

121.  How  many  bags,  each  containing  1  bu.  1  pk.  1  qt. 
1  pt.,  will  be  required  to  hold  184  bu.  5  qt.  of  rye? 

122.  A  meter  is  one  ten-millionth  part  of  a  quarter  of  the 
earth's  circumference.  What  is  the  length  of  the  meter  in 
inches,  if  the  diameter  of  the  earth  is  7912  miles  ? 

123.  Give  the  reason  for  each  step  in  the  operation  of 
finding  the  least  common  multiple  of  3,  7,  9,  12,  14,  18. 

124.  How  many  pounds  of  bread  can  be  made  from  5  bu. 
wheat,  w^eighing  60  lb.  per  bushel,  if  the  wheat  loses  30 
per  cent  during  the  process  of  grinding  into  flour,  and  if  the 
bread  w^eiglis  331  per  cent  more  than  the  weight  of  the 
flour  used? 

125.  A  man  loses  $310.50  on  360  barrels  of  flour  by  sell- 
ing it  at  15  per  cent  below  cost.  What  is  the  selling  imcQ 
per  barrel? 

126.  What  principal  will  amount  to  $715.13  in  2  mo. 
13  da.  at  41  per  cent  ? 

127.  Two  men  start  from  the  same  place,  one  traveling 
north  41  miles  per  hour,  the  other  going  west  at  thS  rate  of 
6  miles  per  hour.  In  what  time  will  they  be  105  miles 
apart  ? 

128.  A  man  owns  a  city  block  600  feet  long,  240  feet 
wdde.  How  many  square  yards  of  flagging  will  be  needed 
to  make  a  sidew^alk  12  feet  wide  surrounding  his  property  ? 

129.  A's  men  can  do  a  piece  of  work  in  27  days,  and  B's 
men  can  do  it  in  36  days.  In  how  many  days  wdll  the  work 
be  done  if  one  half  of  A's  men  and  one  third  of  B's  men  are 
employed  ? 


294  Arithmetic 

130.  A  train  going  at  a  uniform  rate  starts  from  W  at 
7.15  A.M.,  and  arrives  at  X  at  9.45  a.m.  After  a  stop  of  15 
minutes,  it  goes  on  to  Y,  where  it  also  stops  15  minutes,  and 
it  reaches  Z  at  4.45  p.m.  The  distance  between  W  and  X  is 
100  miles,  between  X  and  T  120  miles.  Find  the  time  of 
arrival  at  Y,  and  the  distance  between  Y  and  Z. 

131.  What  per  cent  is  gained  on  beans  bought  at  $2.56 
per  bushel  (2150.4  cu.  in.),  and  sold  at  33  cents  per  gallon, 
liquid  measure  (231  cu.  in.)  ? 

132.  If  a  piece  of  wire  3  rods  2  yards  1  foot  6  inches 
long  cost  $2.28,  what  will  be  the  cost  of  a  piece  19  rods 

4  yards  2  feet  9  inches  long? 

133.  A  vessel  has  sailed  due  east  17°  11'  15".  Find  the 
distance  traveled  in  miles,  the  length  of  a  degree  being 
48.64  miles. 

134.  In  ten  years  the  population  of  a  state  increased 
from  332,286  to  332,422.  Find  the  percentage  of  increase. 
Find  the  percentage  of  increase  in  the  population  of  another 
state  in  the  same  time  from  98,268  to  328,808.  Carry  out 
to  two  places  of  decimals  in  each  case. 

135.  A  note  for  $145.20  was  discounted  at  a  bank  at 

5  per  cent,  and  the  proceeds  amounted  to  $143.99.     For 
how  many  days  was  the  note  discounted  ? 

136.  No  allowance  being  made  for  mortar,  how  many 
bricks  will  be  required  to  build  a  wall  3  rods  long,  2  yards 
high,  and  1  foot  3  inches  thick,  each  brick  being  8  inches 
long,  4  inches  wide,  and  2|-  inches  thick  ? 

137.  At  $80  per  acre,  what  will  be  the  cost  of  a  piece  of 
land  in  the  form  of  a  right-angled  triangle,  the  base  measur- 
ing 120  rods  and  the  perpendicular  measuring  64  rods  ? 
Find  the  cost  of  fencing  it  at  $1.20  per  rod. 

138.  At  $  60  per  M  find  the  cost  of  30  boards,  each  16  ft. 
long,  16  inches  wide. 


Miscellaneous   Problems  295 

139.  Find  the  interest  on  ^1250  for  1  yr.  6  mo.  15  da.  at 

41%. 

140.  A  invests  $1200  and  loses  $300  in  4  months ;  B  in- 
vests $750  and  loses  $250  in  6  months.  Which  loses  the 
greater  per  cent  per  month  on  his  investment,  and  what  per 
cent  greater  is  his  loss  ? 

141.  How  much  greater  income  fractionally  will  A's  in- 
come be  by  buying  5%  stock  at  80  than  by  buying  7%  stock 
at  117  ?     (Brokerage  included  in  })rices  given.) 

142.  Two  men  drive  in  the  same  direction  around  a 
square  two  miles  on  a  side,  starting  from  the  same  point  at 
the  same  time.  The  first  goes  at  the  rate  of  five  miles  per 
hour  and  the  second  at  the  rate  of  six  miles  per  hour.  How 
far  must  the  first  go  before  they  are  again  together  ? 

143.  How  many  gallons  of  vinegar  are  there  in  a  mixture 
of  791  gallons  containing  6%  of  water  ? 

144.  If  the  interest  of  $625  for  3  yr.  7  mo.  6  da.,  at  a 
certain  rate  per  cent,  is  $  135,  in  what  time  will  $  800  pro- 
duce $  76  at  -|  as  great  a  rate  ? 

145.  A  cellar  is  32  feet  long,  8  feet  deep,  and  16  feet 
wide.  What  would  be  the  depth  of  a  cubical  cistern  of 
equal  capacity? 

146.  A  drover  sold  a  lot  of  cattle  at  an  advance  of  20%. 
If  they  had  cost  him  $200,  he  would  have  lost  10%. 
What  did  he  pay  for  the  cattle? 

147.  A  dealer  mixed  45  bushels  of  oats,  37  bushels  of 
corn,  and  43  bushels  of  wheat  for  chicken  feed,  paying  25 
cents  for  the  oats,  50  cents  for  the  corn,  and  75  cents  for 
the  wheat  per  bushel.  What  is  the  cost  of  5  bushels  of  the 
mixture  ? 

148.  Find  the  exact  interest  on  $3650  from  April  14  to 
Sept.  6  at  5%. 

149.  Multiply  6472  by  9612. 


296  Arithmetic 

150.  What  sum  invested  in  8  per  cent  bonds  at  162|  will 
yield  an  income  of  §1200  per  year? 

151.  At  $4  per  cord  what  is  the  value  of  a  pile  of  wood 
18'x8'6"x4'? 

152.  If  a  grain  of  gold  can  be  beaten  out  into  leaves 
covering  56  square  inches,  how  many  square  feet  will  an 
ounce  of  gold  cover  when  beaten  out? 

153.  Multiply  9867  by  6097|f 

154.  I  sold  a  carriage  for  §360  on  which  I  lost  20%; 
what  did  I  lose?  I  sold  a  carriage  for  $360,  on  which  I 
gained  §  72 ;  what  per  cent  did  I  gain  ? 

'    155.    Find  the  interest   on  $1080  for   1  year  6  months 
15  days,  at  5  per  cent. 

156.  A  farmer  sells  a  pile  of  cordwood  ten  feet  long,  six 
feet  high,  and  four  feet  wide  at  $6  a  cord.  How  many 
shingles  at  $5  per  thousand  can  he  obtain  for  the  money 
received  through  the  sale  of  the  wood? 

157.  What  number  is  325%  of  1875? 

158.  State  and  solve  a  problem  that  will  require  the 
operations  indicated  below : 

$5.85x150 
13  X  18  X  2-1- 

159.  Divide  1.736  by  1.6;  1736  by  .16;  .1736  by  16. 

160.  A  man  pays  $324  for  a  piano.  What  is  the  "list 
price,"  if  he  has  been  allowed  discounts  of  40  and  10%  ? 

161.  A  90-day  note  for  $1200  was  discounted  at  a  bank 
June  27.  The  holder  received  $1184,  the  rate  being  6%. 
What  was  the  date  of  the  note  ? 

162.  What  is  the  total  surface  of  a  cube,  one  edge  of 
which  measures  6|-  inches? 

163.  The  divisor  is  973,  the  quotient  8060,  the  remainder 
549  ;  what  is  the  dividend? 


Miscellaneous   Problems  297 

164.  Divide  .63  by  .4.  Explain  full}'  the  reason  for  your 
location  of  the  decimal  point  in  the  qnotient. 

165.  How  many  half-pint  bottles  can  be  filled  from  a 
cask  containing  24  gal.  3  qt.  1  pt.  of  vinegar  ? 

166.  What  number  multiplied  by  2|  and  divided  by  6 J 
equals  3^  ? 

167.  What  is  the  sum  of  7  miles  4  yards,  8  miles  225 
rods,  and  118  rods  3  yards  ?     Give  answer  without  fractions. 

168.  A  and  B  own  a  mill,  A's  share  being  f  and  B's  the 
remainder.  If  A's  share  is  worth  £  4381  2s.  6d.,  what  is 
the  value  in  English  money  of  B's  share  ? 

169.  A  merchant  buys  green  coffee  for  IG^f^  x^er  pound, 
and  pays  |  ^  per  pound  for  roasting.  If  the  coffee  loses 
15%  in  weight  during  the  roasting,  what  price  must  the 
merchant  ask  that  he  may  gain  3  cents  on  each  pound  sold  ? 

170.  A  man's  expenses  are  $1260,  and  he  saves  $540; 
what  per  cent  of  his  income  does  he  spend  ? 

171.  If  railroad  stock  sells  at  160,  including  brokerage, 
what  per  cent  semiannual  dividends  must  be  declared  so 
that  the  stock  may  return  5%  annually  on  the  investment  ? 

172.  Find  the  entire  surface  of  a  square  pyramid,  whose 
altitude  is  144  feet,  one  side  of  the  base  being  34  feet. 

173.  Find  the  weight  of  a  cubic  yard  of  granite,  granite 
being  2.65  times  as  heavy  as  water,  and  a  cubic  foot  of 
water  weighing  1000  ounces. 

174.  What  is  the  value  of  a  carload  of  oats  weighing 
20,000  lb.,  at  42  cents  per  bushel  of  32  pounds  ? 

175.  I  owe  $900  on  Oct.  16,  and  $500  on  Dec.  20.  If 
I  pay  the  former  on  Oct.  1,  15  days  before  it  is  due,  at  what 
date  should  I  be  permitted  to  pay  the  latter  ? 

176.  A  farmer  had  60  acres  in  oats,  twice  as  much  in 
wheat,  and  two  thirds  as  much  in  corn  as  in  wheat.  His  net 
profit  per  acre  was  $  6  on  the  wheat,  which  was  20  %  more 


298  Arithmetic 

than  his  profit  per  acre  on  the  corn,  and  25%  more  than  his 
profit  per  acre  on  the  oats.  How  much  more  would  he  have 
gained  if  he  had  sowed  all  three  parcels  in  wheat? 

177.  The  Julian  calendar  assumed  the  length  of  a  year 
as  365  da.  6  hr.  instead  of  365  da.  5  lir.  48  min.  49.7  sec. 
To  how  many  hours  did  the  difference  amount  in  100 
years  ? 

178.  Four  commercial  travelers  have  routes  which  they 
cover  in  30,  35,  40,  and  45  days,  respectively.  If  they  start 
from  headquarters  the  same  day,  how  many  days  will 
elapse  before  all  meet  again  at  headquarters  ? 

179.  Divide  $1200  among  three  persons  so  that  the 
second  shall  receive  25%  more  than  the  first,  and  the  third 
20%  more  than  the  second. 

180.  If  a  schoolroom  40  feet  by  30  feet  has  8  windows, 
each  having  8  lights  of  glass  20  inches  by  24  inches,  what 
is  the  ratio  of  the  lighting  surface  to  the  floor  surface  ? 

181.  At  27  j^  per  cubic  yard,  find  the  cost  of  excavating  a 
cellar  in  sloping  ground,  the  length  being  108  feet,  the 
width  60  feet,  the  depth  being  9  feet  at  one  end,  and  4|-  feet 
at  the  other. 

182.  The  captain  of  a  ship  makes  an  observation  of  the 
sun  at  noon,  at  which  moment  his  chronometer,  keeping 
Greenwich  time,  indicates  2.15  p.m.  In  what  longitude 
is  the  ship? 

183.  A  fence  8  wires  high  is  put  around  a  square  field 
containing  10  acres.  Each  strand  of  wire  weighs  1  lb.  per 
rod  and  costs  2^^  per  lb.,  and  the  posts,  placed  1  rod  apart, 
cost  15  cents  each.     Find  the  cost  of  the  fence. 

184.  At  $45  per  M,  find  the  cost  of  four  sticks  of  timber 
each  16  ft.  long,  12  inches  wide,  and  10  inches  thick. 

185.  If  4^  buys  an  8-oz.  loaf  when  flour  is  $5  per 
barrel,  how  large  a  loaf  should  be  bought  for  6^  when  flour 
is  $6  per  barrel? 


TABLES 


LINEAR  MEASURE 

12    inches  (in.) =  1  foot .     . 

3    feet =1  yard     . 

^  yards,  or  16 J  feet  .     .     .  =  1  rod  .     . 

40    rods =  1  furlong 

320    rods =  1  mile      . 


ft. 

yd. 

rd. 


fur. 
mi. 


1  mi.  =  320  rd.  =  1760  yd.  =  5280  ft.  =  63,360  iu. 
A  hand,  used  in  measuring  the  height  of  horses,  =  4  in.    A  knot,  used 


m  measuring  distances  at  sea, 
the  depth  of  the  sea,  =  6  ft. 


1.15  mi.    A  fathom,  used  in  measuring 


SQUARE  MEASURE 

144  square  inches  (sq.  in.)  .     .  =  1  square  foot 

9  square  feet =1  square  yard 

30^  sq.  yd.,  or  272i  sq.  ft.   .     .  =  1  square  rod 

160  square  rods =1  acre       .     . 

640  acres =1  square  mile 


sq.  ft. 
sq.  yd. 
sq.  rd. 
A. 
sq.  mi. 


1  A.  =  100  sq.  rd.  =  4840  sq.  yd.  =  43,560  sq.  ft. 

A  Section  of  land  is  a  square  mile. 

Koofing,  flooring,  and  slating  are  often  estimated  by  the  square, 
which  contains  100  square  feet. 


SURVEYORS'  MEASURE 

In  measuring  land,  surveyors  use  a  chain  (ch.)  which  contains  100 
links  (1.)  and  is  4  rods  long.  Since  the  cliain  is  4  rods  long,  a  square 
chain  contains  16  sq.  rd.,  and  10  sq.  ch.  =  160  sq.  rd.,  or  1  acre. 

CUBIC  MEASURE 

1728  cubic  inches  (cu.  in.)  .     .  =  1  cubic  foot      .     .     .  cu.  ft. 

27  cubic  feet =  1  cubic  yard     .     .     .  cu.  yd. 

128  cubic  feet =1  cord cd. 

10  cubic  feet =  1  cord  ft cd.  ft. 

8  cord  feet =  1  cord cd. 

Note.  —  In  computing  the  contents  of  an  enclosing  wall,  masons  and 
brick-layers  regard  it  as  one  straight  wall  whose  length  is  the  distance 
around  it  on  the  outside.    Corners  are  thus  measured  twice. 

A  perch  r2  stone  or  masonry  is  lO^  ft.  long,  1|  ft.  thick,  and  1  ft. 
high,  and  contains  24|  cu.  ft. 


YB  35888 


MEASURES  OF  CAPACITY 

Liquid  Measlke  Dry  Measure 

4  gills      =  1  pint    .     .     .     pt.         2  pints    =  1  quart    .     .     qt. 

2  pints    =  1  quart .     .     .     qt.         8  quarts  =  1  peck     .     .     pk. 

4  quarts  =  1  gallon     .     .    gal.       4  pecks   =  1  bushel .     .     bu. 

The  standard  gallon  contains  231  cubic  inches. 
The  standard  bushel  contains  2150.42  cubic  inches. 

The  capacity  of  cisterns,  reservoirs,  etc.,  is  often  expressed  in  barrels 
(bl)l.)  of  3I5  gallons  each,  or  in  hogsheads  (hhd.)  of  63  gallons  each.  In 
commerce,  these  vary  in  size. 

AVOIRDUPOIS  WEIGHT 

16  ounces  (oz.)      .     .     .  =  1  pound lb. 

100  pounds =  1  hundredweight.     .     cwt. 

2000  pounds    .....=  1  ton T. 


Thi 
and  ii 


1  bus 
1  bus 
1  bus 
1  bus 


54i;{71 


uses 


)lb. 
lib. 
)lb. 
•  lb. 


UNIVERSITY  OF  CAUFORNIA  LIBRARY 


8  drams =  1  ounce  .     .     .     oz.,or;^. 

12  ounces =  1  pound  .     .     .     lb.,  or  lb. 

One  pound  Apothecaries'  weight  =  5700  grains. 

BRITISH  OR  STERLING  MONEY 

4  farthings =1  penny d. 

12  pence =1  shilling s. 

20  shillings =1  pound £. 

5  shillings =  1  crown. 

The  value  of  £1  is  64.8GG5  in  United  States  gold  coin. 

The  unit  of  French  money  is  1  franc,  which  is  19.3  cents.    The  unit  of 
German  money  is  1  mark,  wliich  is  23.85  cents. 


. 

• 

111! 

}! 

I 


